Etale Cohomology Theory

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NANKAI TRACTS IN MATHEMATICS Series Editors: Yiming Long and Weiping Zhang Chern Institute of Mathematics

Published Vol. 1

Vol. 2

Scissors Congruences, Group Homology and Characteristic Classes by J. L. Dupont The Index Theorem and the Heat Equation Method by Y. L.Yu

Vol. 3

Least Action Principle of Crystal Formation of Dense Packing Type and Kepler’s Conjecture by W. Y. Hsiang

Vol. 4

Lectures on Chern–Weil Theory and Witten Deformations by W. P. Zhang

Vol. 5

Contemporary Trends in Algebraic Geometry and Algebraic Topology edited by Shiing-Shen Chern, Lei Fu & Richard Hain

Vol. 6

Riemann–Finsler Geometry by Shiing-Shen Chern & Zhongmin Shen

Vol. 7

Iterated Integrals and Cycles on Algebraic Manifolds by Bruno Harris

Vol. 8

Minimal Submanifolds and Related Topics by Yuanlong Xin

Vol. 9

Introduction to Several Complex Variables: Three Methods of Several Complex Variables by Yum-Tong Siu et al.

Vol. 10 Differential Geometry and Physics edited by Mo-Lin Ge & Weiping Zhang Vol. 11 Inspired by S S Chern edited by Phillip A. Griffiths Vol. 12 Topology and Physics edited by Kevin Lin, Zhenghan Wang & Weiping Zhang Vol. 13 Etale Cohomology Theory by Lei Fu

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Nankai Tracts in Mathematics – Vol. 13

ETALE COHOMOLOGY THEORY

Lei Fu

Chern Institute of Mathematics, Nankai University, China

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

ETALE COHOMOLOGY THEORY Nankai Tracts in Mathematics — Vol. 13 Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-4307-72-7 ISBN-10 981-4307-72-6

Printed in Singapore.

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Preface

The manuscript of this book was written in 1998–1999. At that time, I was invited to give a series of talks at the Morningside Center of Mathematics on Deligne’s proof of the Weil conjecture. To prepare the talks, I wrote the book [Fu (2006)] on algebraic geometry covering the main materials in [EGA] I–III, and the current book covering the main materials in [SGA] 1, 4, 4 12 , and 5 related to etale cohomology theory. I hope this book provides adequate preparation for reading more advanced papers such as [Beilinson, Bernstein and Deligne (1982)], [Deligne (1974)], [Deligne (1980)] and [Laumon (1987)]. The prerequisites for reading this book are [Fu (2006)] and the book [Matsumura (1970)] on commutative algebra. As [Fu (2006)] may not be widely available, whenever a result from it is quoted, a corresponding result in [EGA] or [Hartshorne (1977)] is also indicated. A result used in this book but not covered in these books is Artin’s approximation theorem [Artin (1969)]. A nice account can be found in [Bosch, L¨ utkebohmert and Raynaud (1990)]. At the beginning of each section, I give a list of references related to the content of this section. I strongly encourage the reader to go through these references, especially [SGA], for more general and thorough treatment. When I was a graduate student, the books [Freitag and Kiehl (1988)] and [Milne (1980)] on etale cohomology theory gave me great help for reading [SGA]. It is inevitable that some treatments in this book are influenced by them. During the preparation of this book, I am supported by the Qiu Shi Science & Technologies Foundation and the NSFC. Lei Fu Chern Institute of Mathematics v

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Contents

Preface

v

1.

1

Descent Theory 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

2.

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Etale Morphisms and Smooth Morphisms 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

3.

Flat Modules . . . . . . . . . . . . . Faithfully Flat Modules . . . . . . . Local Criteria for Flatness . . . . . . Constructible Sets . . . . . . . . . . Flat Morphisms . . . . . . . . . . . . Descent of Quasi-coherent Sheaves . Descent of Properties of Morphisms Descent of Schemes . . . . . . . . . . Quasi-finite Morphisms . . . . . . . Passage to Limit . . . . . . . . . . .

The Sheaf of Relative Differentials . . . . . Unramified Morphisms . . . . . . . . . . . . Etale Morphisms . . . . . . . . . . . . . . . Smooth Morphisms . . . . . . . . . . . . . . Jacobian Criterion . . . . . . . . . . . . . . Infinitesimal Liftings of Morphisms . . . . . Direct Limits and Inverse Limits . . . . . . Henselization . . . . . . . . . . . . . . . . . Etale Morphisms between Normal Schemes

Etale Fundamental Groups

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1 3 10 15 18 21 28 35 41 45 59

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3.1 3.2 3.3 4.

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Triangulated Categories . Derived Categories . . . . Derived Functors . . . . . RHom(−, −) and − ⊗L A− Way-out Functors . . . .

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Divisors . . . . . . . . . . . . . . . Cohomology of Curves . . . . . . . Proper Base Change Theorem . . . Cohomology with Proper Support Cohomological Dimension of Rf∗ . Local Acyclicity . . . . . . . . . . . Smooth Base Change Theorem . . Finiteness of Rf! . . . . . . . . . .

171 176 193 201 205 210 222 245 257 267

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Base Change Theorems 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

139 146 152 161 164 171

ˇ Presheaves and Cech Cohomology Etale Sheaves . . . . . . . . . . . . Stalks of Sheaves . . . . . . . . . . Recollement of Sheaves . . . . . . . The Functor f! . . . . . . . . . . . Etale Cohomology . . . . . . . . . Calculation of Etale Cohomology . Constructible Sheaves . . . . . . . Passage to Limit . . . . . . . . . .

Derived Categories and Derived Functors 6.1 6.2 6.3 6.4 6.5

7.

Group Cohomology . . . . . . . . Profinite Groups . . . . . . . . . Cohomology of Profinite Groups Cohomological Dimensions . . . . Galois Cohomology . . . . . . . .

139

Etale Cohomology 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

6.

Finite Group Actions on Schemes . . . . . . . . . . . . . . 117 Etale Covering Spaces and Fundamental Groups . . . . . 121 Functorial Properties of Fundamental Groups . . . . . . . 131

Group Cohomology and Galois Cohomology 4.1 4.2 4.3 4.4 4.5

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267 272 279 287 303 311

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311 317 331 350 368 377 391 406

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ix

Contents

8.

Duality 8.1 8.2 8.3 8.4 8.5 8.6

9.

411

Extensions of Henselian Discrete Valuation Rings Trace Morphisms . . . . . . . . . . . . . . . . . . Duality for Curves . . . . . . . . . . . . . . . . . The Functor Rf ! . . . . . . . . . . . . . . . . . . Poincar´e Duality . . . . . . . . . . . . . . . . . . Cohomology Classes of Algebraic Cycles . . . . .

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Finiteness Theorems 9.1 9.2 9.3 9.4 9.5 9.6

Sheaves with Group Actions . . . . Nearby Cycle and Vanishing Cycle Generic Base Change Theorem and Generic Local Acyclicity . . . . . . Finiteness of RΨη . . . . . . . . . Finiteness Theorems . . . . . . . . Biduality . . . . . . . . . . . . . .

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10. `-adic Cohomology 10.1 10.2 10.3 10.4 10.5

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Adic Formalism . . . . . . . . . . . . . . Grothendieck–Ogg–Shafarevich Formula Frobenius Correspondences . . . . . . . Lefschetz Trace Formula . . . . . . . . . Grothendieck’s Formula of L-functions .

509 517 521 526 531

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531 566 585 592 603

Bibliography

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Index

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Chapter 1

Descent Theory

Unless otherwise stated, rings in this book are commutative with the identity element 1, and homomorphisms of rings map 1 to 1. For any ring A and any A-module M , we assume 1 · x = x for all x ∈ M . For any scheme (S, OS ) and any s ∈ S, denote the maximal ideal pf OS,s by ms , and denote the residue field of OS,s by k(s). For any nonnegative integer n, denote by AnS the affine space Spec OS [t1 , . . . , tn ] over S, and by PnS the projective space Proj OS [t0 , t1 , . . . , tn ] over S. We identify AnS with the open subscheme Spec OS [ tt01 , . . . , ttn0 ] of PnS . 1.1

Flat Modules

([SGA 1] IV 1.) Let A be a ring. An A-module M is called flat if the functor N 7→ M ⊗A N on the category of A-modules is exact. We also say that M is flat over A, or A-flat. Let A → B be a homomorphism of rings. If M is a flat A-module, then B ⊗A M is a flat B-module. If N is a flat B-module, and B is flat over A, then N is flat over A. Proposition 1.1.1. Let A be a ring and M an A-module. The following conditions are equivalent: (i) M is flat. (ii) For any A-modules N , we have TorA i (M, N ) = 0 for all i ≥ 1. (iii) For any finitely generated A-module N , we have TorA i (M, N ) = 0 for all i ≥ 1. (iv) For any A-module N , we have TorA 1 (M, N ) = 0. (v) For any finitely generated A-module N , we have TorA 1 (M, N ) = 0. (vi) For any ideal I of A, we have TorA (M, A/I) = 0. 1 1

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(vii) For any finitely generated ideal I of A, we have TorA 1 (M, A/I) = 0. (viii) For any ideal I of A, the canonical homomorphism I ⊗A M → M,

a ⊗ x 7→ ax

I ⊗A M → M,

a ⊗ x 7→ ax

is injective, that is, it induces an isomorphism I ⊗A M ∼ = IM . (ix) For any finitely generated ideal I of A, the canonical homomorphism

is injective. Let M be a flat A-module, N an A-module, N 0 and N 00 submodules of N . Then M ⊗A N 0 and M ⊗A N 00 can be regarded as submodules of M ⊗A N . We have M ⊗A (N 0 ∩ N 00 ) ∼ = (M ⊗A N 0 ) ∩ (M ⊗A N 00 ), M ⊗A (N 0 + N 00 ) ∼ = (M ⊗A N 0 ) + (M ⊗A N 00 ),

where on the right-hand side, we take the intersection and the summation inside M ⊗A N . Proposition 1.1.2. (i) Let A be a ring and let S be a multiplicative subset in A. Then S −1 A is flat over A. If M is a flat A-module, then S −1 M is a flat S −1 A-module. (ii) Let A → B be a homomorphism of rings, let S (resp. T ) be a multiplicative subset in A (resp. B) such that the image of S in B is contained in T , and let N be a B-module. If N is flat over A, then T −1 N is flat over A and over S −1 A. (iii) Let A → B be a homomorphism of rings and let N be a B-module. Suppose for every maximal ideal n of B, Nn is flat over A. Then N is flat over A. Proof.

Let us prove (ii). For any A-module M , we have T −1 N ⊗A M ∼ = T −1 (N ⊗A M ).

If N is flat over A, the functor T −1 (N ⊗A −) on the category of A-modules is exact. It follows that T −1 N is flat over A. By (i), S −1 T −1 N is flat over S −1 A. We have S −1 T −1 N ∼ = T −1 N . Proposition 1.1.3. (i) Let A be a ring and let M be a flat A-module. If a ∈ A is not a zero divisor, then the canonical homomorphism M → M,

x 7→ ax

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Descent Theory

is injective. In particular, if A is an integral domain, then M has no torsion. (ii) Let A be an integral domain such that Am is a discrete valuation ring for every maximal ideal m of A. Then an A-module M is flat if and only if it has no torsion. Proof. Let us prove the “if” part of (ii). Suppose M has no torsion. To prove M is A-flat, it suffices to show Mm is Am -flat for any maximal ideal m of A. Let I be an ideal of Am . By our assumption, I is principal, say generated by some element r ∈ A. The canonical map Am → I,

a 7→ ra

is an isomorphism. So we have an isomorphism ∼ =

M m → I ⊗ Am M m ,

x 7→ r ⊗ x.

The composite of this isomorphism with the canonical homomorphism I ⊗ Am M m → M m ,

a ⊗ x → ax

is Mm → Mm ,

x 7→ rx,

which is injective since M has no torsion. We then apply 1.1.1 (viii). 1.2

Faithfully Flat Modules

([SGA 1] IV 2–4.) Let C and D be two categories, and let F : C → D be a functor. We say that F is faithful if for all objects X and Y in C , the map HomC (X, Y ) → HomD (F (X), F (Y )),

f 7→ F (f )

is injective. If C and D are additive categories and F is an additive functor, then the above condition is equivalent to saying that the condition F (u) = 0 implies the condition u = 0 for any u ∈ HomC (X, Y ). In this case, the condition F (X) = 0 implies the condition X = 0 for any object X in C . Indeed, we have F (idX ) = idF (X) = 0, and hence idX = 0. A functor F : C → D is called fully faithful if the map HomC (X, Y ) → HomD (F (X), F (Y )),

f 7→ F (f )

is bijective for all objects X, Y ∈ ob C . F is called essentially surjective if for any object Z in D, there exists an object X in C such that F (X) ∼ = Z.

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Etale Cohomology Theory

We say that F is an equivalence of categories if F is fully faithful and essentially surjective. Proposition 1.2.1. Let C and D be abelian categories and let F : C → D be an additive functor. The following conditions are equivalent: (i) F is exact and faithful. (ii) A sequence M 0 → M → M 00 in C is exact if and only if F (M 0 ) → F (M ) → F (M 00 ) is exact. (iii) F is exact and the condition F (X) = 0 implies the condition X = 0. Suppose furthermore that there exists a family of nonzero objects {Zi } in C such that for any nonzero object X in C , there exist some Zi and some object Y in C admitting a monomorphism Y → X and an epimorphism Y → Zi . Then the above conditions are equivalent to the following: (iv) F is exact and F (Zi ) 6= 0 for all Zi . Proof. (i)⇒(ii) Given a sequence u

v

M 0 → M → M 00 in C , suppose F (u)

F (v)

F (M 0 ) → F (M ) → F (M 00 ) is exact. We have F (vu) = F (v)F (u) = 0. Since F is faithful, we have vu = 0. Hence im u ⊂ ker v. Since F is exact, we have F (ker v/im u) ∼ = F (ker v)/F (im u) ∼ = ker F (v)/im F (u) = 0. Hence ker v/im u = 0, that is, ker v = im u. (ii)⇒(iii) If F (X) = 0, then F (0) → F (X) → F (0) is exact. Our condition implies that 0→X →0 is exact. So X = 0. (iii)⇒(i) Let u : X → Y be a morphism in C . If F (u) = 0, then im F (u) = 0. Since F is exact, we have F (im u) ∼ = im F (u) = 0. By our condition, we have im u = 0, that is, u = 0.

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(iii)⇒ (iv) is obvious. (iv)⇒(iii) For any nonzero object X in C , choose an object Y admitting a monomorphism Y → X and an epimorphism Y → Zi . Then F (Y ) → F (Zi ) is an epimorphism. As F (Zi ) 6= 0, we have F (Y ) 6= 0. The morphism F (Y ) → F (X) is a monomorphism. It follows that F (X) 6= 0. Corollary 1.2.2. Let A be a ring and M an A-module. The following conditions are equivalent: (i) The functor N 7→ M ⊗A N on the category of A-modules is exact and faithful. (ii) A sequence of A-modules N 0 → N → N 00 is exact if and only if M ⊗A N 0 → M ⊗A N → M ⊗A N 00 is exact. (iii) M is flat and the condition M ⊗A N = 0 implies the condition N = 0. (iv) M is flat and M ⊗A A/m 6= 0 for any maximal ideal m of A. When M satisfies the above equivalent conditions, we say that M is faithfully flat. Corollary 1.2.3. Let (A, m) → (B, n) be a local homomorphism of local rings and let M be a finitely generated B-module. Then M is faithfully flat over A if and only if it is flat over A and nonzero. Indeed, by Nakayama’s lemma, the condition M ⊗A A/m 6= 0 is equivalent to the condition M 6= 0. Proposition 1.2.4. Let A → B be a homomorphism of rings. If there exists a B-module M faithfully flat over A, then the map Spec B → Spec A is onto. Proof. It suffices to show that for any p ∈ Spec A, the fiber Spec (B ⊗A Ap /pAp ) of the map Spec B → Spec A over p is not empty, or equivalently, B ⊗A Ap /pAp is nonzero. Indeed, since M is faithfully flat over A, M ⊗A Ap /pAp is faithfully flat over Ap /pAp . This implies that M ⊗A Ap /pAp 6= 0. But M ⊗A Ap /pAp is a (B ⊗A Ap /pAp )-module. So B ⊗A Ap /pAp 6= 0.

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Etale Cohomology Theory

Corollary 1.2.5. Let φ : A → B be a homomorphism of rings and let M be a finitely generated B-module flat over A. Suppose Supp M = Spec B. Then for any p ∈ SpecA, and any prime ideal q ∈ Spec B which is minimal among those prime ideals of B containing pB, we have φ−1 (q) = p. In particular, for any minimal prime ideal q of B, φ−1 (q) is a minimal prime ideal of A. Proof. By 1.2.3, Mq is faithfully flat over Aφ−1 (q) . By 1.2.4, the map Spec Bq → Spec Aφ−1 (q) is onto. We have pAφ−1 (q) ∈ Spec Aφ−1 (q) . By the minimality of q, the preimage of pAφ−1 (q) in Spec Bq must be qBq . It follows that φ−1 (q) = p. Proposition 1.2.6. Let φ : A → B be a homomorphism of rings. The following conditions are equivalent: (i) B is faithfully flat over A. (ii) B is flat over A and Spec B → Spec A is onto. (iii) B is flat over A, and for every maximal ideal m of A, there exists a maximal ideal n of B such that φ−1 (n) = m. (iv) B is flat over A, and for any A-module M , the canonical homomorphism M → M ⊗A B,

x 7→ x ⊗ 1

is injective. (v) For every ideal I of A, the canonical homomorphism −1

I ⊗A B → B,

x ⊗ b → bx

is injective and φ (IB) = I. (vi) φ is injective and coker φ is flat over A.

Proof. (i)⇒(ii) follows from 1.2.4. (ii)⇒(iii) Let m be a maximal ideal of A. Suppose Spec B → Spec A is onto. Then there exists a prime ideal q of B such that φ−1 (q) = m. Let n be a maximal ideal of B containing q. Then φ−1 (n) = m. (iii)⇒(i) For any maximal ideal m of A, let n be a maximal ideal of B such that φ−1 (n) = m. We have B ⊗A A/m ∼ = B/mB, and B/mB has a quotient B/n which is nonzero. It follows that B ⊗A A/m 6= 0. We then apply 1.2.2. (i)⇒(iv) Suppose B is faithfully flat over A. To show M → M ⊗A B is injective, it suffices to show that the homomorphism M ⊗A B → M ⊗A B ⊗A B,

x⊗b → x⊗1⊗b

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is injective. Indeed, this homomorphism has a left inverse M ⊗A B ⊗A B → M ⊗A B,

x ⊗ b1 ⊗ b2 → x ⊗ b1 b2 .

(iv)⇒(v) Taking M = A/I, we see that the canonical homomorphism A/I → A/I ⊗A B,

a ¯ 7→ a ¯⊗1

is injective. This is equivalent to saying that the canonical homomorphism A/I → B/IB is injective. Hence φ−1 (IB) = I. By 1.1.1 (viii), the canonical homomorphism I ⊗A B → B is injective. (v)⇒ (iii) By 1.1.1 (viii), B is flat over A. For any maximal ideal m of A, we have φ−1 (mB) = m. In particular, mB is a proper ideal of B. Let n be a maximal ideal of B containing mB. Then we have φ−1 (n) = m. (iv)⇒(vi) In (iv), if we take M = A, we see φ is injective. We thus have a short exact sequence 0 → A → B → coker φ → 0. For any A-module M , we have an exact sequence A 0 → TorA 1 (M, B) → Tor1 (M, coker φ) → M → M ⊗A B.

Since B is flat over A, we have TorA 1 (M, B) = 0. As M → M ⊗A B is injective, the above exact sequence shows that TorA 1 (M, coker φ) = 0. By 1.1.1 (iv), coker φ is flat over A. (vi)⇒(iv) We have a short exact sequence 0 → A → B → coker φ → 0. For any A-module M , we have an exact sequence A 0 → TorA 1 (M, B) → Tor1 (M, coker φ) → M → M ⊗A B.

Since coker φ is flat over A, we have TorA 1 (M, coker φ) = 0. The above exact sequence then shows that M → M ⊗A B is injective and TorA 1 (M, B) = 0. Proposition 1.2.7. Let A be a noetherian ring, I an ideal of A, and Aˆ the I-adic completion of A. Then Aˆ is flat over A. It is faithfully flat over A if and only if I is contained in the radical of A. See [Fu (2006)] 2.1.23, or [Matsumura (1970)] (23.L) Corollary 1, and (24.A) Theorem 56. Proposition 1.2.8. Let A be a ring, I an ideal of A, and M an A-module. Suppose one of the following conditions holds:

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Etale Cohomology Theory

(a) I is nilpotent. (b) A is noetherian, I is contained in the radical of A, and M is finitely generated. Then the following conditions are equivalent: (i) M is free over A. (ii) M ⊗A A/I is free over A/I and TorA 1 (M, A/I) = 0. (iii) M ⊗A A/I is free over A/I and the canonical homomorphism ∞ ∞ M M M/IM ⊗A/I I n /I n+1 → I n M/I n+1 M n=0

n=0

is an isomorphism.

Proof. (i)⇒(ii) and (i)⇒(iii) are obvious. (ii)⇒(i) Let xλ (λ ∈ Λ) be a family of elements in M such that their images in M ⊗A A/I form a basis. By Nakayama’s lemma, {xλ } generates M . Let L be a free A-module with rank |Λ|, let L → M be an epimorphism mapping a basis of L to {xλ }, and let R be its kernel. We have a short exact sequence 0 → R → L → M → 0.

Since TorA 1 (M, A/I) = 0, the sequence

0 → R ⊗A A/I → L ⊗A A/I → M ⊗A A/I → 0 is exact. The homomorphism L ⊗A A/I → M ⊗A A/I is an isomorphism by the definition of the homomorphism L → M . So R ⊗A A/I = 0, that is, R/IR = 0. This implies that R = 0 by Nakayama’s lemma. Hence L ∼ =M and M is free. (iii)⇒(i) Keep the notation above. We have a commutative diagram L ∞ ∞ L L/IL ⊗A/I I n /I n+1 → I n L/I n+1 L n=0

M/IM ⊗A/I

↓ L ∞

n=0

n=0

I n /I n+1 →

∞ L

↓

I n M/I n+1 M.

n=0

By the definition of the homomorphism L → M , we have L/IL ∼ = M/IM , and hence the first vertical arrow is an isomorphism. The two horizontal arrows are isomorphisms by our assumption and the fact that L is free. It follows that ∞ ∞ M M I n L/I n+1 L → I n M/I n+1 M n=0

n=0

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is an isomorphism. This implies that R ⊂ ∞ T

n=0

etale

∞ T

I n L. Since L is free and

n=0

I n = 0 under the condition (a) or (b), we have R = 0. Hence L ∼ =M

and M is free.

Corollary 1.2.9. Let m be a maximal ideal of a ring A and let M be an A-module. Suppose one of the following conditions holds: (a) m is nilpotent. (b) A is noetherian and local, and M is finitely generated. Then the following conditions are equivalent: (i) M is free. (ii) M is projective. (iii) M is flat. (iv) TorA 1 (M, A/m) = 0. (v) The canonical homomorphism ∞ ∞ M M M/mM ⊗A/m mn /mn+1 → mn M/mn+1 M n=0

n=0

is an isomorphism. If (A, m) is a local noetherian integral domain with residue field k and fraction field K, and M is finitely generated, then the above conditions are equivalent to (vi) dimK (M ⊗A K) = dimk (M ⊗A k).

Proof. The equivalence of (i)–(v) follows directly from 1.2.8. (i)⇒(vi) is obvious. Suppose (vi) holds. Choose xi ∈ M (i = 1, . . . , n) such that their images in M/mM ∼ = M ⊗A k form a basis. By Nakayama’s lemma, {xi } generates M . Let L be a free A-module of rank n, let L → M be an epimorphism mapping a basis of L to {xi }, and let R be the kernel of L → M . Then we have an exact sequence 0 → R ⊗A K → L ⊗A K → M ⊗A K → 0. Since M ⊗A K has the same dimension as L ⊗A K, we have R ⊗A K = 0. But R is contained in the free A-module L, and hence has no torsion. It follows that R = 0. So L ∼ = M and M is free.

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Local Criteria for Flatness

([SGA 1] IV 5.) Proposition 1.3.1. Let A be a ring, I be an ideal of A, and M an An module. If TorA 1 (M, A/I ) = 0 for any n ≥ 0, then the canonical homomorphism ∞ ∞ M M M/IM ⊗A/I I n /I n+1 → I n M/I n+1 M n=0

n=0

is an isomorphism. The converse is true if I is nilpotent.

n Proof. Suppose TorA 1 (M, A/I ) = 0 for any n ≥ 0. Then for each n, we have an exact sequence

0 → M ⊗A I n /I n+1 → M ⊗A A/I n+1 → M ⊗A A/I n → 0. The homomorphism M ⊗A A/I n+1 → M ⊗A A/I n

can be identified with the canonical homomorphism M/I n+1 M → M/I n M,

and the kernel of the latter is I n M/I n+1 M . It follows that the canonical homomorphism M ⊗A I n /I n+1 → I n M/I n+1 M

is an isomorphism. But Hence

M ⊗A I n /I n+1 ∼ = M/IM ⊗A/I I n /I n+1 .

M/IM ⊗A/I I n /I n+1 ∼ = I n M/I n+1 M. In general, we have an exact sequence n+1 n TorA ) → TorA 1 (M, A/I 1 (M, A/I ) →

M ⊗A I n /I n+1 → M ⊗A A/I n+1 → M ⊗A A/I n → 0.

If the homomorphism

M/IM ⊗A/I I n /I n+1 → I n M/I n+1 M

is an isomorphism, then the homomorphism

M ⊗A I n /I n+1 → M ⊗A A/I n+1

is injective, and hence the homomorphism

n+1 n TorA ) → TorA 1 (M, A/I 1 (M, A/I )

n+1 is onto. If I is nilpotent, then TorA ) = 0 for large n. This 1 (M, A/I A n implies that Tor1 (M, A/I ) = 0 for all n ≥ 0.

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Proposition 1.3.2. Let A → B be a homomorphism of rings and let M be an A-module. The following conditions are equivalent: (i) For every B-module N , we have TorA 1 (M, N ) = 0. (ii) M ⊗A B is flat over B, and TorA (M, B) = 0. 1 Proof. (i)⇒(ii) Taking N = B, we see TorA 1 (M, B) = 0. Let 0 → N 0 → N → N 00 → 0 00 be an exact sequence of B-modules. Since TorA 1 (M, N ) = 0, the sequence

0 → M ⊗A N 0 → M ⊗A N → M ⊗A N 00 → 0 is exact, that is, the sequence 0 → (M ⊗A B) ⊗B N 0 → (M ⊗A B) ⊗B N → (M ⊗A B) ⊗B N 00 → 0 is exact. Hence M ⊗A B is flat over B. (ii)⇒(i) Let L → N be an epimorphism such that L is a free B-module, and let R be the kernel of this epimorphism. We have an exact sequence A TorA 1 (M, L) → Tor1 (M, N ) → M ⊗A R → M ⊗A L → M ⊗A N → 0. A Since TorA 1 (M, B) = 0 and L is free over B, we have Tor1 (M, L) = 0. Since M ⊗A B is flat over B, the homomorphism

M ⊗A R → M ⊗A L is injective. So we have TorA 1 (M, N ) = 0.

Corollary 1.3.3. Let A be a ring, I an ideal of A, and M an A-module. The following conditions are equivalent: (i) TorA 1 (M, N ) = 0 for any A/I-module N . (ii) M ⊗A A/I is flat over A/I, and TorA 1 (M, A/I) = 0. A (iii) Tor1 (M, N ) = 0 for any A-module N annihilated by some power of I. If these conditions are satisfied, then the canonical homomorphism M/IM ⊗A/I is an isomorphism.

∞ M n=0

∞ M I n /I n+1 → I n M/I n+1 M n=0

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Proof. (i)⇔(ii) follows from 1.3.2. (iii)⇒(i) is obvious. n n+1 (i)⇒(iii) We have TorA N ) = 0 for any n ≥ 0. 1 (M, I N/I induction on n and using the long exact sequence for Tor, we n+1 TorA N ) = 0 for all n ≥ 0. Taking n sufficiently large, 1 (M, N/I A get Tor1 (M, N ) = 0. The last assertion follows from 1.3.1.

By see we

Proposition 1.3.4. Let A be a ring, I and ideal of A, and M an A-module. Consider the following conditions: (i) M is flat over A. (ii) M ⊗A A/I is flat over A/I and TorA 1 (M, A/I) = 0. (iii) M ⊗A A/I is flat over A/I and the canonical homomorphism ∞ ∞ M M M/IM ⊗A/I I n /I n+1 → I n M/I n+1 M n=0

n=0

is an isomorphism. (iv) For any n ≥ 0, M ⊗A A/I n is flat over A/I n . We have (i)⇒(ii)⇒(iii)⇒(iv). Suppose that either I is nilpotent, or A is noetherian and M ⊗A N is separated with respect to the I-adic topology for any finitely generated A-module N . Then (iv)⇒(i). Proof. (i)⇒(ii) is trivial. (ii)⇒(iii) follows from 1.3.3. A/I n (iii)⇒(iv) By 1.3.1, we have Tor1 (M/I n M, A/I) = 0. The condition n 1.3.3 (ii) holds if we take A to be nA/I and M to be M/I n M . It follows A/I from 1.3.3 (iii) that we have Tor1 (M/I n M, N ) = 0 for any A/I n -module N . Hence M/I n M is flat over A/I n . If I is nilpotent, then taking n sufficiently large, we see (iv)⇒(i). Suppose A is noetherian and M ⊗A N is separated with respect to the I-adic topology for any finitely generated A-module N . Let us prove M is flat over A under the condition (iv). We need to prove that the canonical homomorphism idM ⊗ i : M ⊗A N 0 → M ⊗A N

is injective for any monomorphism i : N 0 ,→ N between finitely generated A-modules. It suffices to show that ker(idM ⊗ i) is contained in I n (M ⊗A N 0 ) = im (M ⊗A I n N 0 → M ⊗A N 0 )

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for all n ≥ 0, or equivalently, that ker(idM ⊗ i) is contained in

im (M ⊗A V → M ⊗A N 0 ) = ker (M ⊗A N 0 → M ⊗A N 0 /V ),

where V goes over a family of submodules of N 0 which form a base of neighborhoods of N 0 at 0 with respect to the I-adic topology. By the Artin–Rees theorem ([Matsumura (1970)] (11.C) Theorem 15), we can take V = N 0 ∩ I n N (n ≥ 0). We have a commutative diagram M ⊗A N 0 → M ⊗A (N 0 /N 0 ∩ I n N ) ↓ ↓ M ⊗A N → M ⊗A (N/I n N ).

Since the canonical homomorphism N 0 /N 0 ∩ I n N → N/I n N is injective and M ⊗A A/I n is flat over A/I n , the right vertical arrow is injective. This implies that ker(idM ⊗ i) is contained in ker (M ⊗A N 0 → M ⊗A (N 0 /N 0 ∩ I n N )) for all n. Our assertion follows. Theorem 1.3.5. Let A → B be a homomorphism of noetherian rings, I an ideal of A such that IB is contained in the radical of B, and M a finitely generated B-module. The following conditions are equivalent: (i) M is flat over A. (ii) M ⊗A A/I is flat over A/I and TorA 1 (M, A/I) = 0. (iii) M ⊗A A/I is flat over A/I and the canonical homomorphism ∞ ∞ M M M/IM ⊗A/I I n /I n+1 → I n M/I n+1 M n=0

n=0

is an isomorphism. (iv) M ⊗A A/I n is flat over A/I n for all n ≥ 0.

Proof. We use 1.3.4. Let us check that M ⊗A N is separated with respect to the I-adic topology for any finitely generated A-module N . Note that M ⊗A N is a finitely generated B-module. Since IB is contained in the radical of B, M ⊗A N is separated with respect to the IB-adic topology by [Fu (2006)] 1.5.8, or [Matsumura (1970)] (11.D) Corollary 1. Our assertion follows. Proposition 1.3.6. Let (A, m) → (B, n) be a local homomorphism between noetherian local rings, and let u : M 0 → M be a homomorphism between finitely generated B-modules. Suppose M is flat over A. The following conditions are equivalent: (i) u is injective, and coker u is flat over A. (ii) u ⊗ id : M 0 ⊗A A/m → M ⊗A A/m is injective.

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Proof. (i)⇒(ii) Since u is injective, we have an exact sequence 0 → M 0 → M → coker u → 0. Since coker u is flat over A, the sequence 0 → M 0 ⊗A A/m → M ⊗A A/m → coker u ⊗A A/m → 0 is exact. So u ⊗ id is injective. (ii)⇒(i) Consider the commutative diagram L L ∞ ∞ M 0 /mM 0 ⊗A/m mn /mn+1 → M/mM ⊗A/m mn /mn+1 n=0

∞ L

n=0

↓

mn M 0 /mn+1 M 0

n=0

→

∞ L

↓

mn M/mn+1 M.

n=0

Suppose u⊗id is injective. Then the top horizontal arrow is injective. Since M is flat over A, the right vertical arrow is an isomorphism by 1.3.1. One can check the left vertical arrow is surjective. These facts imply that the bottom horizontal arrow is injective, which then implies that u is injective. We thus have a short exact sequence 0 → M 0 → M → coker u → 0. It gives rise to an exact sequence u⊗id

A 0 TorA 1 (M, A/m) → Tor1 (coker u, A/m) → M ⊗A A/m → M ⊗A A/m.

Since M is flat over A, we have TorA 1 (M, A/m) = 0. Since u⊗id is injective, the above exact sequence shows that TorA 1 (coker u, A/m) = 0. By 1.3.5, coker u is flat over A. Proposition 1.3.7. Let (C, l) → (A, m) → (B, n) be local homomorphisms of noetherian local rings, and let M be a finitely generated B-module. Suppose A is flat over C. The following conditions are equivalent: (i) M is flat over A. (ii) M is flat over C and M ⊗C C/l is flat over A ⊗C C/l. Proof. (i)⇒(ii) is obvious.

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(ii)⇒(i) Suppose M is flat over C, and M ⊗C C/l is flat over A ⊗C C/l, that is, M/lM is flat over A/lA. By 1.3.5, to prove M is flat over A, it suffices to show the canonical homomorphism ∞ ∞ M M M/lM ⊗A/lA ln A/ln+1 A → ln M/ln+1 M n=0

n=0

is an isomorphism. Since A and M are flat over C, the canonical homomorphisms ∞ ∞ M M A/lA ⊗C/l ln /ln+1 → ln A/ln+1 A, n=0

M/lM ⊗C/l

∞ M

n=0

n=0

∞ M

ln /ln+1 →

ln M/ln+1 M

n=0

are isomorphisms by 1.3.1. We thus have the following isomorphisms ∞ M M/lM ⊗A/lA ln A/ln+1 A n=0

∼ = M/lM ⊗A/lA A/lA ⊗C/l ∼ = M/lM ⊗C/l ∼ =

∞ M

∞ M

n=0

ln /ln+1

ln M/ln+1 M.

n=0

Our assertion follows.

1.4

n=0

ln /ln+1

∞ M

Constructible Sets

([EGA] 0 9.1–9.2, IV 1.8–1.10.) Let X be a topological space. A subset of X is called locally closed if it is the intersection of an open subset and a closed subset. Suppose X is a noetherian topological space. We say that a subset of X is constructible if it is a union of finitely many locally closed subsets. If X1 and X2 are constructible subsets, then X1 ∪X2 , X1 ∩X2 and X1 −X2 are constructible. Proposition 1.4.1. Let X be a noetherian topological space. A subset Y of X is constructible if and only if for any irreducible closed subset F of X such that Y ∩ F is dense in F , the set Y ∩ F contains a nonempty open subset of F .

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Proof. (⇒) Suppose Y is constructible, F is irreducible and closed such that Y ∩ F is dense in F . Note that Y ∩ F is constructible. Write Y ∩ F = (U1 ∩ F1 ) ∪ · · · ∪ (Un ∩ Fn ),

where each Ui (resp. Fi ) is open (resp. closed). Replacing Fi by Fi ∩ F , we may assume Fi ⊂ F . We have F = Y ∩ F = (U1 ∩ F1 ) ∪ · · · ∪ (Un ∩ Fn ).

Since F is irreducible, we have F = Ui ∩ Fi for some i. But Ui ∩ Fi ⊂ Fi . These facts imply that Fi = F , and Y ∩ F contains the nonempty open subset Ui ∩ Fi of F . (⇐) We use noetherian induction. Suppose that for any irreducible closed subset F of X such that Y ∩ F is dense in F , Y ∩ F contains a nonempty open subset of F . If Y is not constructible, then the set S = {Z|Ø 6= Z ⊂ X, Z is closed, and Y ∩ Z is not constructible}

is nonempty. Since X is a noetherian topological space, S contains a minimal element, say Z0 . If Z0 is not irreducible, we can write Z0 = Z1 ∪Z2 , where Z1 and Z2 are proper closed subsets of Z0 . By the minimality of Z0 , we have Z1 , Z2 6∈ S . Hence Y ∩ Z1 and Y ∩ Z2 are constructible. But then Y ∩ Z0 = (Y ∩ Z1 ) ∪ (Y ∩ Z2 ) is constructible. This contradicts Z0 ∈ S . So Z0 must be irreducible. If Y ∩ Z0 is not dense in Z0 , then by the minimality of Z0 , Y ∩ (Y ∩ Z0 ) is constructible. But Y ∩ (Y ∩ Z0 ) = Y ∩ Z0 . This contradicts Z0 ∈ S . So Y ∩ Z0 is dense in Z0 . By our assumption, there exists a nonempty open subset U of Z0 contained in Y ∩ Z0 . By the minimality of Z0 , Y ∩ (Z0 − U ) is constructible. We have Y ∩ Z0 = U ∪ (Y ∩ (Z0 − U )). This shows that Y ∩ Z0 is constructible. Again this contradicts Z0 ∈ S . So Y must be constructible. Theorem 1.4.2 (Chevalley). Let f : X → Y be a morphism of finite type between noetherian schemes. Then f (X) is constructible. Proof. Cover Y by finitely many affine open subsets Vi and cover each f −1 (Vi ) by finitely many affine open subsets Uij . We have f (X) = ∪i,j f (Uij ). It suffices to show that each f (Uij ) is a constructible subset of Vi . Replacing X by Uij and Y by Vi , we are reduced to the case where X = Spec B and Y = Spec A. Any irreducible closed subset of Spec A is of the form V (p) = {p0 ∈ Spec A|p ⊂ p0 } for some prime idea p of A. The set V (p) can be identified with Spec A/p, and f (Spec B) ∩ V (p) can be identified with the image of the morphism Spec B/pB → Spec A/p obtained

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from f by base change. The closure of the image of this morphism consists of those prime ideals of A/p containing the kernel of the homomorphism A/p → B/pB. If f (Spec B) ∩ V (p) is dense in V (p), then the homomorphism A/p → B/pB is injective. By 1.4.1, to prove the theorem, it suffices to prove the following statement: Let A be a noetherian integral domain, and let A → B be an injective homomorphism such that B is a finitely generated A-algebra. Then there exists a nonzero element a ∈ A such that every prime ideal of A lying in the set D(a) = {p ∈ Spec A|a 6∈ p} is the inverse image of a prime ideal of B. Let us prove this statement. Write B = A[x1 , . . . , xn ], where x1 , . . . , xr are algebraically independent over A and xj (r + 1 ≤ j ≤ n) are algebraic over A[x1 , . . . , xr ]. For each r + 1 ≤ j ≤ n, find a relation d

d −1

pj0 xj j + pj1 xj j

+ · · · + pjdj = 0,

where pji ∈ A[x1 , . . . , xr ] (0 ≤ i ≤ dj ), dj ≥ 1, and pj0 6= 0. Then

n Q

pj0

j=r+1

is a nonzero polynomial in x1 , . . . , xr with coefficients in A. Let a be one of the nonzero coefficients of this polynomial. For any prime ideal p of A with a 6∈ p, let p0 = p[x1 , . . . , xr ], which is a prime ideal of A[x1 , . . . , xr ]. n Q Then pj0 6∈ p0 . So Bp0 is integral over A[x1 , . . . , xr ]p0 . Hence there j=r+1

exists a prime ideal q of B whose inverse image in A[x1 , . . . , xr ] is p0 . The inverse image of q in A is then p.

Corollary 1.4.3. Let f : X → Y be a morphism of finite type between noetherian schemes. If Z is a constructible subset of X, then f (Z) is constructible. Proof. Z is a union of finitely many locally closed subset. It suffices to consider the case where Z is locally closed. Then there exists a subscheme structure on Z, and we can apply 1.4.2 to the morphism Z → Y . Let X be a topological space and let x, y ∈ X. We say that y is a generalization of x, and that x is a specialization of y if x ∈ {y}. Proposition 1.4.4. Suppose X is a noetherian topological space such that every irreducible closed subset has a generic point. Let U be a constructible subset of X and let x be a point in U . Then U contains an open neighborhood of x if and only if every generalization y of x lies in U . Proof. The “only if” part is clear. Let us prove the “if” part. We use noetherian induction. Suppose U contains every generalization of x. If U

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does not contain any open neighborhood of x, then the set S = {Z|x ∈ Z ⊂ X, Z is closed, U ∩ Z is not a neighborhood of x in Z} is not empty. Since X is noetherian, S contains a minimal element, say Z0 . If Z0 is not irreducible, we can write Z0 = Z1 ∪ Z2 , where Z1 and Z2 are proper closed subsets of Z0 . If x ∈ Z1 ∩ Z2 , then by the minimality of Z0 , we can find open neighborhoods Vi (i = 1, 2) of x such that Vi ∩ Zi ⊂ U ∩ Zi . Then U ∩ Z0 contains the neighborhood (V1 ∩ V2 ) ∩ Z0 of x in Z0 . This contradicts Z0 ∈ S . If x ∈ Z1 − Z2 , then we can find an open neighborhood V1 of x such that V1 ∩ Z1 ⊂ U ∩ Z1 . But then U ∩ Z0 contains the neighborhood (V1 − Z2 ) ∩ Z0 of x in Z0 . This contradicts Z0 ∈ S . Similarly, if x ∈ Z2 − Z1 , we are again led to contradiction. So Z0 must be irreducible. Let y be the generic point of Z0 . Then y is a generalization of x. By our assumption, we have y ∈ U . But then U ∩ Z0 is dense in Z0 . By 1.4.1, U ∩ Z0 contains a nonempty open subset V of Z0 . Since Z0 6∈ S , we have x 6∈ V . By the minimality of Z0 , we have Z0 − V ∈ S . So there exists an open neighborhood W of x in X such that W ∩ (Z0 − V ) ⊂ U ∩ (Z0 − V ). But then W ∩ Z0 is neighborhood of x in Z0 contained in U ∩ Z0 . This contradicts Z0 ∈ S . So U contains an open neighborhood of x. Corollary 1.4.5. Let X be a noetherian topological space such that every irreducible closed subset has a generic point. A subset U of X is open if and only if the following two conditions hold: (a) U contains generalizations of points in U . (b) For every x ∈ U , U ∩ {x} contains a nonempty open subset of {x}. Proof. The “only if” part is clear. Let us prove the “if” part. Let F be an irreducible closed subset of X. Then F = {x} for some x ∈ X. If U ∩ F is dense in F , then x ∈ U by condition (a). But then U ∩ F contains a nonempty open subset of F by condition (b). So U is constructible by 1.4.1. By condition (a) and 1.4.4, U is open. One can also study constructible sets on non-noetherian schemes. Confer [EGA] 0 9.1–9.2 and IV 1.8–1.10. 1.5

Flat Morphisms

Proposition 1.5.1. Let f : X → Y be a morphism of finite type between noetherian schemes, x ∈ X, and y = f (x). Then the following conditions are equivalent:

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(i) f maps every neighborhood of x to a neighborhood of y. (ii) For any generalization y 0 of y, there exists a generalization x0 of x such that f (x0 ) = y 0 . (iii) The morphism Spec OX,x → Spec OY,y induced by f is surjective on the underlying topological spaces. Proof. (i)⇒(ii) Suppose y ∈ {y 0 }. Let F be the union of those irreducible components of f −1 ({y 0 }) not containing x. Then X − F is a neighborhood of x. So f (X − F ) is a neighborhood of y, and hence y 0 ∈ f (X − F ). Let x1 ∈ X − F such that f (x1 ) = y 0 . Then x1 lies in an irreducible component C of f −1 ({y 0 }) containing x. Let x0 be the generic point of C. Then x0 is a generalization of x. We have x1 ∈ {x0 }. Hence f (x1 ) ∈ {f (x0 )}, that is, y 0 ∈ {f (x0 )}. On the other hand, we have x0 ∈ C ⊂ f −1 ({y 0 }), and hence f (x0 ) ∈ {y 0 }. Therefore f (x0 ) = y 0 (ii)⇒(i) Let U be an open neighborhood of x in X. Then U is constructible. By 1.4.3, f (U ) is a constructible subset in Y . The condition (ii) implies that f (U ) contains every generalization of y. By 1.4.4, f (U ) is a neighborhood of y. (ii)⇔(iii) is clear. Theorem 1.5.2. Let f : X → Y be a morphism of finite type between noetherian schemes and let F be a coherent OX -module with Supp F = X. Suppose Fx is flat over OY,f (x) for any x ∈ X. Then f is an open mapping. Proof. For any x ∈ X, since Fx is flat over OY,f (x) and Fx 6= 0, it is faithfully flat over OY,f (x) by 1.2.3. By 1.2.4, the morphism Spec OX,x → Spec OY,y induced by f is surjective. So by 1.5.1, f is an open mapping. Lemma 1.5.3. Let A be a noetherian integral domain, B a finitely generated A-algebra, and M a finitely generated B-module. Then there exists a nonzero element f ∈ A such that Mf is free over Af . For a proof, see [Matsumura (1970)] (22.A) Lemma 1. Lemma 1.5.4. Let A be a noetherian ring, B a finitely generated Aalgebra, M a finitely generated B-module, q a prime ideal of B, and p the inverse image of q in A. Suppose Mq is flat over Ap . Then there exists g ∈ B − q such that (M/pM )g is flat over A/p and TorA 1 (M, A/p)g = 0. Proof.

By 1.5.3, there exists f ∈ A − p such that (M/pM )f is flat over

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A/p. Since Mq is flat over A, we have A ∼ TorA 1 (M, A/p)q = Tor1 (Mq , A/p) = 0.

From the exact sequence 0 → TorA 1 (M, A/p) → M ⊗A p → M → M ⊗A A/p → 0,

we deduce that TorA 1 (M, A/p) is a finitely generated B-module. So there 0 exists g 0 ∈ B − q such that TorA 1 (M, A/p)g0 = 0. We then take g = f g . Lemma 1.5.5. Under the condition of 1.5.4, for every prime ideal q0 of B containing q but not containing g, Mq0 is flat over A. Proof. We apply 1.3.5 to the homomorphism A → Bq0 , the ideal I = p of A, and the Bq0 -module Mq0 . Theorem 1.5.6. Let f : X → Y be a morphism of finite type between noetherian schemes and let F be a coherent OX -module. (i) If Y is integral, then there exists a nonempty open subset V in Y such that for any x ∈ f −1 (V ), Fx is flat over OY,f (x) . (ii) In general, the set U of points x ∈ X such that Fx are flat over OY,f (x) is open. Proof. (i) follows from 1.5.3. (ii) We may assume that X = Spec B and Y = Spec A are affine, and F = M ∼ for some finitely generated B-module M . By 1.1.2 (ii), the set U contains generalizations of points in U . Let x ∈ U . By 1.5.5, U ∩ {x} contains a nonempty open subset of {x}. By 1.4.5, U is open. A morphism f : X → Y of schemes is called flat at x ∈ X if OX,x is flat over OY,f (x) . It is called a flat morphism if it is flat at every point of X. It is called a faithfully flat morphism if it is flat and surjective on the underlying topological spaces. If f is a morphism of finite type and X and Y are noetherian schemes, then the set of points in X, where f is flat, is open by 1.5.6. If f is a flat morphism of finite type between noetherian schemes, then f is an open mapping by 1.5.2.

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1.6

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([SGA 1] VIII 1.) A morphism of schemes f : X → Y is called quasi-compact if the inverse image in X of any quasi-compact open subset of Y is quasi-compact. This is equivalent to saying that there exists a covering {Ui } of Y by affine open subsets such that each f −1 (Ui ) is a union of finitely many affine open subsets in X. Let g1 , g2 : B → C be maps of sets. We define g1

ker(B ⇒ C) = {b ∈ B|g1 (b) = g2 (b)}. g2

Let f : A → B be another map of set. We say the sequence f

g1

A→B⇒C g2

g1

is exact if f is injective and im f = ker(B ⇒ C). g2

Theorem 1.6.1. Let g : S 0 → S be a quasi-compact faithfully flat morphism, and let S 00 = S 0 ×S S 0 . For any 1 ≤ i ≤ 2, let pi : S 00 = S 0 ×S S 0 → S 0

be the projection to the i-th factor, and for any 1 ≤ i < j ≤ 3, let pij : S 0 ×S S 0 ×S S 0 → S 00 = S 0 ×S S 0

be the projection to the (i, j)-factor. (i) Let F and G be quasi-coherent OS -modules, and let F 0 and G 0 (resp. 00 F and G 00 ) be their inverse images on S 0 (resp. S 00 ), respectively. Then the sequence g∗

p∗ 1

HomOS (F , G ) → HomOS0 (F 0 , G 0 ) ⇒ HomOS00 (F 00 , G 00 ) p∗ 2

is exact, that is, if u0 ∈ HomOS0 (F 0 , G 0 ) satisfies p∗1 (u) = p∗2 (u), then there exists a unique u ∈ HomOS (F , G ) such that g ∗ u = u0 . (ii) For any 1 ≤ i ≤ 3, let πi : S 0 ×S S 0 ×S S 0 → S 0

be the projection to the i-th factor. Let F 0 be a quasi-coherent sheaf on S 0 ∼ = such that there exists an isomorphism σ : p∗1 F 0 → p∗2 F 0 satisfying p∗13 (σ) = p∗23 (σ) ◦ p∗12 (σ).

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Here we identify p∗13 (σ) (resp. p∗23 (σ), resp. p∗12 (σ)) with a morphism π1∗ F 0 → π3∗ F 0 (resp. π2∗ F 0 → π3∗ F 0 , resp. π1∗ F 0 → π2∗ F 0 ) through the canonical natural transformations p∗13 ◦ p∗1 ∼ = π1∗ , p∗ ◦ p∗ ∼ = π∗ , 23

1

2

p∗12 ◦ p∗1 ∼ = π1∗ ,

p∗13 ◦ p∗2 ∼ = π3∗ , p∗ ◦ p∗ ∼ = π∗ , 23

2

3

p∗12 ◦ p∗2 ∼ = π2∗ .

Then there exist a quasi-coherent OS -module F and an isomorphism τ : ∼ = g ∗ F → F 0 such that the diagram p∗ τ

1 p∗1 g ∗ F → p∗1 F 0 ∼ ∼ =↓ =↓ σ

p∗ τ

2 p∗2 g ∗ F → p∗2 F 0

commutes, where the left vertical arrow is the canonical isomorphism. Moreover F is unique up to unique isomorphism. Any isomorphism σ : p∗1 F 0 → p∗2 F 0 satisfying p∗13 (σ) = p∗23 (σ) ◦ p∗12 (σ) is called a descent datum for F 0 . Proof. We leave it for the reader to reduce the proof of the theorem to the case where S and S 0 are affine. So we assume S = Spec A, S 0 = Spec A0 , and A0 is a faithfully flat A-algebra. (i) Let M and N be A-modules such that F = M ∼ and G = N ∼ , respectively. Set M 0 = M ⊗A A0 ,

M 00 = M ⊗A A0 ⊗A A0 ,

N 0 = N ⊗A A0 ,

N 00 = N ⊗A A0 ⊗A A0 .

We need to show that the sequence HomA (M, N ) → HomA0 (M 0 , N 0 ) ⇒ HomA00 (M 00 , N 00 ) is exact. By 1.2.6 (iv), N → N 0 is injective. It follows that HomA (M, N ) → HomA0 (M 0 , N 0 )

is injective. Let u0 ∈ ker (HomA0 (M 0 , N 0 ) ⇒ HomA00 (M 00 , N 00 )).

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To show that u0 ∈ im (HomA (M, N ) → HomA0 (M 0 , N 0 )), it suffices to show that u0 maps the A-submodule M of M 0 to the Asubmodule N of N 0 . For any x ∈ M , u0 (x) lies in ker (N 0 ⇒ N 00 ). So it suffices to prove N → N 0 ⇒ N 00 is exact, that is, φ

ψ

0 → N → N ⊗A A0 → N ⊗A A0 ⊗A A00 is exact, where φ(x) = x ⊗ 1,

ψ(x ⊗ a0 ) = x ⊗ a0 ⊗ 1 − x ⊗ 1 ⊗ a0

for any x ∈ N and a0 ∈ A0 . Since A0 is faithfully flat over A, it suffices to prove the sequence φ⊗id

ψ⊗id

0 → N ⊗A A0 → N ⊗A A0 ⊗A A0 → N ⊗A A0 ⊗A A0 ⊗A A0 is exact. Consider the diagram 0 → N ⊗A A0 . ↓ id

0 → N ⊗A A0

φ⊗id

→ N ⊗A A0 ⊗A A0 D1 . ↓ id φ⊗id

→ N ⊗A A0 ⊗A A0

ψ⊗id

→ N ⊗A A0 ⊗A A0 ⊗A A0 , D2 . ψ⊗id

→ N ⊗A A0 ⊗A A0 ⊗A A0 ,

where D1 (x ⊗ a1 ⊗ a2 ) = x ⊗ a1 a2 ,

D2 (x ⊗ a1 ⊗ a2 ⊗ a3 ) = x ⊗ a1 ⊗ a2 a3

for any x ∈ N and ai ∈ A0 (i = 1, 2, 3). Then D1 ◦ (φ ⊗ id) = id,

(φ ⊗ id) ◦ D1 + D2 ◦ (ψ ⊗ id) = id,

that is, Di (i = 1, 2) define a homotopy between the identity and the zero chain map. Using these identities, one can verify that the sequence is exact. (ii) The uniqueness of F follows from (i). Assume F 0 = N 0∼ for some ∼ = 0 A -module N 0 . The isomorphism σ : p∗1 F 0 → p∗2 F 0 is induced by an isomorphism ∼ =

α : N 0 ⊗A A0 → A0 ⊗A N 0 of A0 ⊗A A0 -modules. Set N = {x ∈ N 0 |α(x ⊗ 1) = 1 ⊗ x}.

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Then N is an A-module and we have an exact sequence γ

N → N 0 ⇒ A0 ⊗A N 0 , δ

where γ(x0 ) = α(x0 ⊗ 1), δ(x0 ) = 1 ⊗ x0 for any x0 ∈ N 0 . Since A0 is faithfully flat over A, the sequence γ⊗id

N ⊗A A0 → N 0 ⊗A A0 ⇒ A0 ⊗A N 0 ⊗A A0 δ⊗id

is exact. The homomorphism A0 → A0 ⊗A A0 , a0 7→ 1 ⊗ a0 0 0 makes A ⊗A A a faithfully flat A0 -algebra. By the proof of (i), we have an exact sequence N 0 → N 0 ⊗A0 (A0 ⊗A A0 ) ⇒ N 0 ⊗A0 ((A0 ⊗A A0 ) ⊗A0 (A0 ⊗A A0 )), that is, ξ

N 0 → A0 ⊗A N 0 ⇒ A0 ⊗A A0 ⊗A N 0 η

is exact, where ξ(a0 ⊗ x0 ) = a0 ⊗ 1 ⊗ x0 , η(a0 ⊗ x0 ) = 1 ⊗ a0 ⊗ x0 0 for any a ∈ A0 and x0 ∈ N 0 . Consider the diagram γ⊗id

N ⊗A A0 → N 0 ⊗A A0 ⇒ A0 ⊗A N 0 ⊗A A0 δ⊗id

β

↓

N0

α

↓

p∗ 23 (α) ξ

↓

→ A0 ⊗A N 0 ⇒ A0 ⊗A A0 ⊗A N 0 . η

Using the assumption p∗13 (σ) = p∗23 (σ) ◦ p∗12 (σ), one checks that the squares on the right-hand side commute. So there exists a homomorphism of A0 modules β : N ⊗A A0 → N 0 making the square on the left-hand side commute. It induces an isomor∼ = phism τ : g ∗ F → F 0 , where F = N ∼ . By the commutativity of the diagram above, we have 1 ⊗ β(x ⊗ 1) = α(x ⊗ 1) for any x ∈ N . But α(x ⊗ 1) = 1 ⊗ x for any x ∈ N . So 1 ⊗ β(x ⊗ 1) = 1 ⊗ x. The images of 1 ⊗ β(x ⊗ 1) and 1 ⊗ x under the map A0 ⊗A N 0 → N 0 , a0 ⊗ x0 → a0 x0 are β(x ⊗ 1) and x, respectively. So β(x ⊗ 1) = x for any x ∈ N . Using this fact, one checks that σ ◦ p∗1 τ is identified with p∗2 τ .

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Corollary 1.6.2. Let g : S 0 → S be a quasi-compact faithfully flat morphism, let h : S 0 ×S S 0 → S 00 be the canonical morphism, and let G be a quasi-coherent OS -module. Then the sequence G → g ∗ g ∗ G ⇒ h∗ h∗ G is exact, where the two morphisms g∗ g ∗ G ⇒ h∗ h∗ G are the canonical morphisms g∗ g ∗ G → g∗ pi∗ p∗i g ∗ G ∼ = h∗ h∗ G

(i = 1, 2)

induced by the two projections p1 , p2 : S 0 ×S S 0 → S 0 . Proof.

For any open subset U of S, the sequence HomOU (OU , G |U ) → HomOg−1 (U ) (Og−1 (U) , g ∗ G |g−1 (U) ) ⇒ HomOh−1 (U ) (Oh−1 (U) , h∗ G |h−1 (U) )

is exact by 1.6.1, that is, the sequence G (U ) → (g∗ g ∗ G )(U ) ⇒ (h∗ h∗ G )(U ) is exact.

Corollary 1.6.3. Let g : S 0 → S be a quasi-compact faithfully flat morphism, let F be a quasi-coherent OS -module, and let F 0 (resp. F 00 ) be the inverse image of F on S 0 (resp. S 0 ×S S 0 ). Denote by Quot(F ) (resp. Quot(F 0 ), resp. Quot(F 00 )) the set of isomorphic classes of quasi-coherent quotient sheaves of F (resp. F 0 , resp. F 00 ). Then the sequence g∗

0

p∗ 1

Quot(F ) → Quot(F ) ⇒ Quot(F 00 ) p∗ 2

is exact. Proof. Let G 0 ∈ ker(Quot(F 0 ) ⇒ Quot(F 00 )). Then we have an isomor∼ = phism σ : p∗1 G 0 → p∗2 G 0 such that the diagram p∗1 F 0 → p∗1 G 0 ∼ =↓ F 00 ↓σ ∼ =↓ p∗2 F 0 → p∗2 G 0

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commutes. So the following diagrams commute, where F 000 is the inverse image of F on S 0 ×S S 0 ×S S 0 : p∗12 p∗1 F 0 → p∗12 p∗1 G 0 ∼ = ↓ F 000 ↓p∗12 (σ) ∼ = ↓ p∗12 p∗2 F 0 → p∗12 p∗2 G 0 ,

p∗23 p∗1 F 0 → p∗23 p∗1 G 0 ∼ = ↓ F 000 ↓p∗23 (σ) ∼ = ↓ p∗23 p∗2 F 0 → p∗23 p∗2 G 0 ,

p∗13 p∗1 F 0 → p∗13 p∗1 G 0 ∼ = ↓ F 000 ↓p∗13 (σ) ∼ = ↓ p∗13 p∗2 F 0 → p∗13 p∗2 G 0 .

Since the horizontal arrows are surjective, this implies that p∗13 (σ) = p∗23 (σ) ◦ p∗12 (σ). By 1.6.1 (ii), there exists a quasi-coherent OS -module G whose inverse image on S 0 is isomorphic to G 0 . By 1.6.1 (i), there exists a morphism F → G inducing the epimorphism F 0 → G 0 . Since g : S 0 → S is faithfully flat, F → G is surjective. So G is a quotient of F . The uniqueness of G follows from 1.6.1 (i). Corollary 1.6.4. Let g : S 0 → S be a quasi-compact faithfully flat morphism, let F be a quasi-coherent OS -module, and let F 0 (resp. F 00 ) be the inverse image of F on S 0 (resp. S 0 ×S S 0 ). Denote by Sub(F ) (resp. Sub(F 0 ), resp. Sub(F 00 )) the set of isomorphic classes of quasi-coherent subsheaves of F (resp. F 0 , resp. F 00 ). Then the sequence g∗

p∗ 1

Sub(F ) → Sub(F 0 ) ⇒ Sub(F 00 ) p∗ 2

is exact. We leave the proof to the reader. Corollary 1.6.5. Let g : S 0 → S be a quasi-compact faithfully flat morphism, let S 00 = S 0 ×S S 0 , and let Sub(S) (resp. Sub(S 0 ), resp. Sub(S 00 )) be the set of isomorphic classes of closed subschemes of S (resp. S 0 , resp. S 00 ). Then the sequence g∗

p∗ 1

Sub(S) → Sub(S 0 ) ⇒ Sub(S 00 ) p∗ 2

is exact, where g ∗ , p∗i (i = 1, 2) denote base changes of closed subschemes. Proof.

This follows from 1.6.3 or 1.6.4 by taking F = OS .

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Proposition 1.6.6. Let g : S 0 → S be a quasi-compact faithfully flat morphism, and let F be a quasi-coherent OS -module. Then g ∗ F is locally of finite type (resp. locally of finite presentation, resp. locally free of finite rank) if and only if F is so. Proof. The “if” part is clear. Let us prove the “only if” part. We leave it to the reader to reduce the proof to the case where S = Spec A and S 0 = Spec A0 are affine and A0 is a faithfully flat A-algebra. Let M be an A-module. Write M = limλ Mλ , where Mλ goes over the set of finitely −→ generated submodules of M . We have M ⊗A A0 = limλ Mλ ⊗A A0 . Suppose −→ M ⊗A A0 is a finitely generated A0 -module. Then there exists some Mλ such that Mλ ⊗A A0 → M ⊗A A0 is surjective. Since A0 is faithfully flat over A, we must have Mλ = M . So M is a finitely generated A-module. If M ⊗A A0 has finite presentation, then M must be a finitely generated A-module. So we can find an exact sequence 0→R→L→M →0 such that L is a free A-module of finite rank. As A0 is faithfully flat over A, the sequence 0 → R ⊗A A0 → L ⊗A A0 → M ⊗A A0 → 0 is exact. Since M ⊗A A0 has finite presentation, R ⊗A A0 is a finitely generated A0 -module. This implies that R is a finitely generated A-module. So M has finite presentation. The statement about local freeness follows from 1.6.7 below and the fact that M is flat over A if and only if M ⊗A A0 is flat over A0 . Lemma 1.6.7. Let M be an A-module. The following conditions are equivalent: (i) M ∼ is a locally free OSpec A -module of finite rank. (ii) M is a finitely generated projective A-module. (iii) M is a flat A-module of finite presentation. Proof. (i)⇒(ii) Since M ∼ is locally free of finite rank, we can find a finite affine open covering {Ui } of Spec A such that M ∼ |Ui are free of finite rank. The ` canonical morphism g : i Ui → S is quasi-compact and faithfully flat, and g ∗ M ∼ has finite presentation. So M has finite presentation by 1.6.6. For any A-module N , we then have H omOX (M ∼ , N ∼ ) ∼ = (HomA (M, N ))∼ .

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Since M ∼ is locally free, the functor H omOX (M ∼ , −) is exact on the category of OSpec A -modules. So the functor HomA (M, −) is exact on the category of A-modules. Hence M is a projective A-module. (ii)⇒(iii) There exists an A-module R such that An ∼ = M ⊕ R for some finite n. It follows that M is flat and has finite presentation. (iii)⇒(i) For any p ∈ Spec A, let us prove M ∼ is free in a neighborhood of p. Mp is a flat Ap -module with finite presentation. This implies that Mp is a free Ap -module of finite rank. (Confer the proof of (ii)⇒(i) of 1.2.8.) Choose x1 , . . . , xk ∈ M so that their images in Mp form a basis. Consider the homomorphism φ : Ak → M which maps a basis of Ak to {x1 , . . . , xk }. We have (ker φ)p = (coker φ)p = 0. Since M is finitely generated, so it is with coker φ. It follows that there exists an affine open neighborhood of p in Spec A on which (coker φ)∼ vanishes. Replacing Spec A by this neighborhood, we may assume coker φ = 0. Then φ is an epimorphism. Since M has finite presentation, ker φ is finitely generated. There exists an affine open neighborhood of p in Spec A on which (ker φ)∼ vanishes. Replacing Spec A by this neighborhood, we may assume ker φ = 0. Then φ is an isomorphism, and M ∼ is free.

1.7

Descent of Properties of Morphisms

([SGA 1] VIII 3, 4) Let f : X → S be a morphism of schemes. We say that f is surjective (resp. injective) if f is surjective (resp. injective) on the underlying topological spaces. f is called radiciel if it is universally injective, that is, for any morphism S 0 → S, the base change f 0 : X ×S S 0 → S 0 of f is injective. Proposition 1.7.1. Let f : X → S be a morphism. The following conditions are equivalent: (i) f is radiciel. (ii) For any algebraically closed field K, the map X(K) → S(K) induced by f is injective, where X(K) = Hom(Spec K, X) and S(K) = Hom(Spec K, S) are the sets K-points in X and S, respectively. (iii) f is injective, and for any x ∈ X, the residue field k(x) of X at x is a purely inseparable algebraic extension of the residue field k(f (x)) of Y at f (x). (iv) The diagonal morphism ∆ : X → X ×S X is surjective.

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In particular, any radiciel morphism is separated. Proof. (i)⇒(ii) Suppose f is universally injective and t1 , t2 : Spec K → X are two morphisms such that f t1 = f t2 . Let us prove t1 = t2 . Regard Spec K as an S-scheme through the morphism f t1 = f t2 . Let Γti : Spec K → X ×S Spec K

(i = 1, 2)

be the graphs of ti , and let fK : X ×S Spec K → Spec K be the base change of f . It suffices to show Γt1 = Γt2 . We have fK Γt1 = fK Γt2 = idSpec K . Since fK is injective, Γt1 and Γt2 have the same image in X ×Y Spec K. Let x0 be their common image. Then the composites f\

Γ\t

K k(x0 ) →i K K→

(i = 1, 2)

\ are the identity, where fK and Γ\ti are the homomorphisms on residue fields induced by the morphisms fK and Γti , respectively. It follows that \ , and hence they Γ\ti (i = 1, 2) are isomorphisms, and are inverse to fK coincide. So we have Γt1 = Γt2 . (ii)⇒(i) Let g : S 0 → S be a morphism, and let x01 and x02 be two points in X ×S S 0 having the same image s0 in S 0 . The homomorphism k(s0 ) → k(x01 ) is injective. Since k(x02 ) is flat over k(s0 ), the canonical homomorphism

k(x02 ) → k(x01 ) ⊗k(s0 ) k(x02 ) is injective. It follows that 0 6= 1 in k(x01 ) ⊗k(s0 ) k(x02 ). So k(x01 ) ⊗k(s0 ) k(x02 ) has a maximal ideal m. Let K be an algebraic closure of (k(x01 ) ⊗k(s0 ) k(x02 ))/m. Then we have a commutative diagram k(s0 ) → k(x01 ) ↓ ↓ k(x02 ) → K. For each i ∈ {1, 2}, the homomorphism k(x0i ) → K defines a morphism si : Spec K → X×S S 0 having the image x0i in X×S S 0 . Let g 0 : X×S S 0 → X and f 0 : X ×S S 0 → S 0 be the projections. We have f 0 s1 = f 0 s2 . It follows that f g 0 s1 = gf 0 s1 = gf 0 s2 = f g 0 s2 .

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By our assumption, we then have g 0 s1 = g 0 s2 . Together with the identity f 0 s1 = f 0 s2 , this implies that s1 = s2 . In particular, we have x01 = x02 . So f is universally injective. (i)⇔(iii) Suppose f is injective. Let x be a point in X and let s be its image in S. Then f −1 (s) consists of only one point x. Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f g S 0 → S. For any point s0 in S 0 with image s in S, we have a one-to-one correspondence between the underlying set of f 0−1 (s0 ) and the underlying set of the scheme Spec (k(x) ⊗k(s) k(s0 )). It follows that f is universally injective if and only if for any field k 0 containing k(s), the scheme Spec (k(x) ⊗k(s) k 0 ) contains only one point. Let us prove that this last condition is equivalent to saying that k(x) is a purely inseparable algebraic extension of k(s). For convenience, we denote k(s) by k, and k(x) by K. If K is not algebraic over k, then we can find an element t in K which is transcendental over k. We have k(t) ⊗k k[T ] ∼ = k(t)[T ]. Let m = (T − t) be the prime ideal of k(t)[T ] generated by T − t. By the transcendence of t, we have m ∩ k[T ] = 0. On the other hand, the zero prime ideal p of k(t)[T ] also has the property p ∩ k[T ] = 0. It follows that there are at least two points in the fiber of Spec k(t)[T ] → Spec k[T ] over the zero prime ideal of k[T ]. Hence Spec (k(t) ⊗k k(T )) contains at least two points. Since Spec (K ⊗k k(T )) → Spec (k(t) ⊗k k(T )) is surjective, Spec (K ⊗k k(T )) contains at least two points. If K is algebraic over k, but not pure inseparable over k, then we can find a subfield F of K strictly containing k and separable over k. Let k¯ ¯ contains more than one be an algebraic closure of k. Then Spec (F ⊗k k) point. As ¯ → Spec (F ⊗k k) ¯ Spec (K ⊗k k)

¯ contains more than one point. is surjective, Spec (K ⊗k k) If K is a purely inseparable algebraic extension of k, then for any t ∈ K, n we have tp ∈ k for a large integer n, where p is the characteristic of k.

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For any field k 0 containing k, and any t0 ∈ K ⊗k k 0 , we have t0p ∈ k 0 for a large integer n. Let p be a prime ideal of K ⊗k k 0 . We have p ∩ k 0 = 0. For n n any t0 ∈ p, suppose t0p ∈ k 0 . Then t0p ∈ p ∩ k 0 = 0. So any prime ideal of K ⊗k k 0 is nilpotent. This implies that K ⊗k k 0 has only one prime ideal. (ii)⇒(iv) Given a point z in X ×S X, let K be an algebraic closure of k(z). We then have a morphism t : Spec K → X ×S X with image z. Let p1 , p2 : X ×S X → X be the projections. We have f p1 t = f p2 t. So p1 t = p2 t by our assumption. Let t0 = p1 t = p2 t. Then we have pi (∆t0 ) = (pi ∆)t0 = t0 = pi t

(i = 1, 2).

So we have ∆t0 = t. In particular, z lies in the image of ∆. So ∆ is surjective. (iv)⇒(ii) Let t1 , t2 : Spec K → X be two morphisms such that f t1 = f t2 . Then there exists a morphism t : Spec K → X ×S X such that pi t = ti for i = 1, 2. Let z be the image of t. Since ∆ is surjective, there exists a point x ∈ X such that ∆(x) = z. Since ∆ is an immersion, it induces an ∼ = isomorphism ∆\ : k(z) → k(x). So there exists a morphism t0 : Spec K → X 0 such that ∆t = t. But then ti = pi t = pi ∆t0 = t0

(i = 1, 2).

So we have t1 = t2 .

Proposition 1.7.2. Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S → S. Assume g is surjective. (i) f is surjective if and only if f 0 is so. (ii) If f 0 is injective, then so is f . (iii) f is radiciel if and only if f 0 is so. Proof.

For any s0 ∈ S 0 , we have

f 0−1 (s0 ) ∼ = f −1 (g(s0 )) ⊗k(g(s0 )) k(s0 ).

The projection f 0−1 (s0 ) → f −1 (g(s0 )) is surjective by 1.2.4 and the fact that k(s0 ) is faithfully flat over k(g(s0 )). So f 0−1 (s0 ) 6= Ø if and only if f −1 (g(s0 )) 6= Ø. Since g is surjective, this implies (i). If f 0 is injective, then f 0−1 (s0 ) contains at most one point. So f −1 (g(s0 )) contains at most one point, and hence f is injective. This shows (ii). (iii) follows from (ii).

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Proposition 1.7.3. Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S → S. Assume g is quasi-compact and surjective. Then f is quasi-compact if and only if f 0 is so. Proof. Suppose f is quasi-compact. To prove that f 0 is quasi-compact, it suffices to show that for any affine open subset V 0 of S 0 such that g(V 0 ) is contained in an affine open subset V of S, f 0−1 (V 0 ) is a union of finitely Sn many affine open subsets of X ×S S 0 . We can write f −1 (V ) = i=1 Ui for finitely many affine open subsets Ui in X. Then we have f 0−1 (V 0 ) =

n [

i=1 0

(V 0 ×V Ui ),

where each V ×V Ui is regarded as an affine open subscheme of X ×S S 0 . Conversely suppose f 0 is quasi-compact. Let V be a quasi-compact open subset of S. Since g is surjective, we have V = g(g −1 (V )). So f −1 (V ) = f −1 (g(g −1 (V )) = g 0 (f 0−1 g −1 (V )). Since g and f 0 are quasi-compact, f 0−1 g −1 (V ) is quasi-compact, and hence f −1 (V ) = g 0 (f 0−1 g −1 (V )) is quasi-compact. So f is quasi-compact. Proposition 1.7.4. Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S → S. Assume g is quasi-compact and faithfully flat. Then f is of finite type if and only if so is f 0 . The proof is similar to that of 1.6.6. Proposition 1.7.5. Let g : Y 0 → Y be a flat morphism and let Z be a subset of Y . Assume there exists a quasi-compact morphism f : X → Y such that Z = f (X). Then g −1 (Z) = g −1 (Z). Proof. The inclusion g −1 (Z) ⊂ g −1 (Z) is clear. Let y 0 ∈ g −1 (Z). We need to show y 0 ∈ g −1 (Z). Choose an affine open neighborhood U 0 = Spec A0 of y 0 and an affine open neighborhood U = Spec A of g(y 0 ) such

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that g(U 0 ) ⊂ U . We have y 0 ∈ U 0 ∩ g −1 (Z ∩ U ∩ U ). To prove y 0 ∈ g −1 (Z), it suffices to show y 0 ∈ g −1 (Z ∩ U ) ∩ U 0 . Moreover, Z ∩ U is the image of the quasi-compact morphism f −1 (U ) → U induced from f . We are thus reduced to the case where Y 0 = Spec A0 and Y = Spec A are affine and A0 is a flat A-algebra. Since f : X → Y is quasi-compact and Y is now affine, we can cover X by finitely many open affine subsets Vi . Replacing X by ` i Vi , we may assume X = Spec B for some A-algebra B. Let I be the kernel of A → B. We have g −1 (Z) = g −1 (f (X)) = g −1 (V (I)) = V (IA0 ).

Let f 0 : X 0 = Spec (A0 ⊗A B) → Y 0 = Spec A0

be the base change of f . Since A0 is flat over A, the kernel of A0 → A0 ⊗A B is IA0 . It follows that f 0 (X 0 ) = V (IA0 ). So we have g −1 (Z) = g −1 (f (X)) = f 0 (X 0 ) = V (IA0 ). We thus have g −1 (Z) = g −1 (Z). This proves our assertion.

Corollary 1.7.6. Let g : Y 0 → Y be a quasi-compact flat morphism, and let Z 0 be a closed subset of Y 0 satisfying Z 0 = g −1 (g(Z 0 )). Then Z 0 = g −1 (g(Z 0 )). Moreover, the subspace topology on g(Y 0 ) induced from Y coincides with the quotient topology induced from Y 0 . Proof. The first assertion follows from 1.7.5. To prove the second assertion, note that since the morphism g : Y 0 → Y is continuous, every subset of g(Y 0 ) that is closed with respect to the subspace topology induced from Y is closed with respect to the quotient topology induced from Y 0 . Conversely, let Z be a subset of g(Y 0 ) closed with respect to the quotient topology induced from Y 0 . Then g −1 (Z) is closed. Applying the first assertion to Z 0 = g −1 (Z), we get g −1 (Z) = g −1 (g(g −1 (Z))), that is, g −1 (Z) = g −1 (Z). Hence Z = Z ∩ g(Y 0 ). So Z is closed with respect to the topology induced from Y . Corollary 1.7.7. Assume g : Y 0 → Y is a quasi-compact faithfully flat morphism. Then the topology on Y coincides with the quotient topology induced from Y 0 . Corollary 1.7.8. Let g : S 0 → S be a quasi-compact faithfully flat morphism, S 00 = S 0 ×S S 0 , O(S) (resp. O(S 0 ), resp. O(S 00 )) the set of open subsets of S (resp. S 0 , resp. S 00 ), and F(S) (resp. F(S 0 ), resp. F(S 00 )) the

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set of closed subsets of S (resp. S 0 , resp. S 00 ). Then the following sequences are exact: p−1 1

O(S) → O(S 0 ) ⇒ O(S 00 ), p−1 2

p−1 1

F(S) → F(S 0 ) ⇒ F(S 00 ). p−1 2

Corollary 1.7.9. Assume g : Y 0 → Y is a quasi-compact faithfully flat morphism. Let Z be a subset of Y which is the image of a quasi-compact morphism. (For example, Y is noetherian and Z is constructible.) Then Z is locally closed if and only if g −1 (Z) is so. Proof.

Consider the Cartesian diagram g −1 (Z) → Z ↓ ↓ g Y 0 → Y,

wherein we put a closed subscheme structure on Z. The morphism g −1 (Z) → Z is quasi-compact and faithfully flat. By 1.7.5, we have g −1 (Z) = g −1 (Z). If g −1 (Z) is locally closed, then it is open in g −1 (Z). By 1.7.7, Z is open in Z. So Z is locally closed. Corollary 1.7.10. Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S → S.

Assume g is quasi-compact and faithfully flat. If f 0 is an open mapping (resp. a closed mapping, resp. a quasi-compact embedding, resp. a homeomorphism), then f is so. Proof. For any subset Z of X, we have g −1 (f (Z)) = f 0 (g 0−1 (Z)). If Z is open (resp. closed) and f 0 is an open mapping (resp. a closed mapping), then f 0 g 0−1 (Z) is open (resp. closed). So g −1 (f (Z)) is open (resp. closed). By 1.7.7, f (Z) is open (resp. closed). So f is an open (resp. closed) mapping. Suppose f 0 is a quasi-compact embedding. Then f is injective by 1.7.2, and quasi-compact by 1.7.3. Let Z be a closed subset of X. By 1.7.5, we have g −1 (f (Z)) = g −1 (f (Z)) = f 0 (g 0−1 (Z)). Since f 0 is an embedding, we have f 0−1 (f 0 (g 0−1 (Z))) = g 0−1 (Z).

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It follows that g 0−1 (f −1 (f (Z))) = f 0−1 (g −1 (f (Z))) = f 0−1 (f 0 (g 0−1 (Z))) = g 0−1 (Z). Since g 0 is surjective, this implies that f −1 (f (Z)) = Z. Hence f is an embedding. If f 0 is a homeomorphism, then f is an embedding by what we have proved, and f is surjective by 1.7.2. So f is a homeomorphism. Corollary 1.7.11. Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S → S. Assume g is quasi-compact and faithfully flat. Then f is universally open, universally closed, or universally a homeomorphism, if and only if f 0 is so. Corollary 1.7.12. Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S → S. Assume g is quasi-compact and faithfully flat. Then f is separated or proper if and only if so is f 0 . Proof. f is separated if and only if the diagonal morphism X → X ×S X is a closed mapping. f is proper if it is of finite type, separated, and universally closed. Our assertions follow from 1.7.10, 1.7.4, and 1.7.11. 1.8

Descent of Schemes

([SGA 1] VIII 2, 5.) Proposition 1.8.1. Let g : S 0 → S be a morphism of schemes. (i) Suppose g is surjective and OS → g∗ OS 0 is injective. Then g is an epimorphism in the category of schemes. (ii) Suppose g is surjective and the topology on S is the quotient topology induced from S 0 . Let S 00 = S 0 ×S S 0 , let p1 , p2 : S 00 = S 0 ×S S 0 → S 0 be the projections, and let h : S 00 → S be the canonical morphism. Assume p\1

OS → g∗ OS 0 ⇒ h∗ OS 00 p\2

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is exact. Then for any scheme Z, the sequence p∗ 1

g∗

Hom(S, Z) → Hom(S 0 , Z) ⇒ Hom(S 00 , Z) p∗ 2

is exact, where g ∗ maps a morphism f : S → Z to f g, and p∗i (i = 1, 2) map a morphism f : S 0 → Z to f pi : S 00 → Z. Proof. We leave the proof of (i) to the reader. Let us prove (ii). By (i), the map g ∗ : Hom(S, Z) → Hom(S 0 , Z) is injective. Let f 0 : S 0 → Z be a morphism of schemes such that f 0 p1 = f 0 p2 . For any s01 , s02 ∈ S 0 such that g(s01 ) = g(s02 ), there exists s00 ∈ S 00 such that pi (s00 ) = s0i (i = 1, 2). We then have f 0 (s01 ) = f 0 p1 (s00 ) = f 0 p2 (s00 ) = f 0 (s02 ). As g is surjective, there exists a map f : S → Z on the underlying set such that f g = f 0 . Since the topology on S is the quotient topology induced from S 0 , f is continuous. We have a commutative diagram f∗0 OS 0 → f∗0 p1∗ OS 00 % ok ok p\1

OZ → f∗ g∗ OS 0 ⇒ f∗ h∗ OS 00 p\2

& 0

ok

f∗0 OS 0

0

→

ok

f∗0 p2∗ OS 00 .

Since f p1 = f p2 , the image of the morphism OZ → f∗ g∗ OS 0 lies in the kernel of f∗ g∗ OS 0 ⇒ f∗ h∗ OS 00 . By our assumption, we have an exact sequence f∗ OS → f∗ g∗ OS 0 ⇒ f∗ h∗ OS 00 . So the morphism OZ → f∗ g∗ OS 0 induces a morphism OZ → f∗ OS . The pair formed by the continuous map f : S → Z and the morphism OZ → f∗ OS of sheaves of rings defines a morphism f : S → Z of ringed spaces such that f 0 = f g. For any s0 ∈ S 0 , the composite of the homomorphisms \ fg(s 0)

g \0

s OZ,f g(s0 ) → OS,g(s0 ) → OS 0 ,s0

coincides with the homomorphism fs0\0 : OZ,f g(s0 ) → OS 0 ,s0 . Since f 0 and g are morphisms of schemes, fs0\0 and gs\0 are local homomorphisms of local \ rings. It follows that fg(s 0 ) is also a local homomorphism. Hence f is a morphism of schemes.

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Corollary 1.8.2. Let g : S 0 → S be a quasi-compact faithfully flat morphism, and let S 00 = S 0 ×S S 0 . For any scheme Z, the sequence Hom(S, Z) → Hom(S 0 , Z) ⇒ Hom(S 00 , Z) is exact. Proof.

Use 1.6.2, 1.7.7, and 1.8.1.

Corollary 1.8.3. Let g : S 0 → S be a quasi-compact faithfully flat morphism, S 00 = S 0 ×S S 0 , X and Y two S-schemes, X 0 = X ×S S 0 , Y 0 = Y ×S S 0 , X 00 = X ×S S 00 , and Y 00 = Y ×S S 00 . Then the sequence HomS (X, Y ) → HomS 0 (X 0 , Y 0 ) ⇒ HomS 00 (X 00 , Y 00 ) is exact. Proof.

By 1.8.2, the sequences Hom(X, Y ) → Hom(X 0 , Y ) ⇒ Hom(X 00 , Y ) Hom(X, S) → Hom(X 0 , S) ⇒ Hom(X 00 , S)

are exact. It follows that the sequence HomS (X, Y ) → HomS (X 0 , Y ) ⇒ HomS (X 00 , Y ) is exact. The above sequence can be identified with the sequence in the corollary. Corollary 1.8.4. In the notation of 1.8.3, let f : X → Y be an Smorphism, and let f 0 : X 0 → Y 0 be the base change of f . Then f is an isomorphism if and only if so is f 0 . Proof. Suppose f 0 is an isomorphism. Let g 0 be its inverse. Then g 0 lies in the kernel of HomS 0 (Y 0 , X 0 ) ⇒ HomS 00 (Y 00 , X 00 ) since its images under these two maps are both the inverse of the base change f 00 : X 00 → Y 00 of f . Thus there exists an S-morphism g : Y → X so that g 0 is its base change. Since g 0 f 0 = idX 0 and the map HomS (X, X) → HomS 0 (X 0 , X 0 ) is injective, we have gf = idX . Similarly, we have f g = idY . So f is an isomorphism. Corollary 1.8.5. Under the assumption of 1.8.4, f is a closed immersion, an open immersion, or a quasi-compact immersion, if and only if f 0 is so.

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Proof. Suppose f 0 is a closed immersion. By 1.6.5, there exists a closed ∼ = immersion f0 : X0 → Y such that we have an isomorphism φ0 : X 0 → X00 0 0 0 0 0 0 0 with the property that f0 φ = f , where f0 : X0 = X0 ×S S → Y is the base change of f0 . Let p∗i (φ0 ) (i = 1, 2) be the images of the φ0 under the maps HomY 0 (X 0 , X00 ) ⇒ HomY 00 (X 00 , X000 ). We have f000 p∗1 (φ0 ) = f 00 = f000 p∗2 (φ0 ), where f 00 and f000 are the base changes of f and f0 , respectively. Since f000 is a closed immersion, this implies that p∗1 (φ0 ) = p∗2 (φ0 ). By 1.8.3, there exists a morphism φ : X → X0 such that φ0 is its base change and f0 φ = f . By 1.8.4, φ is an isomorphism. As f0 is a closed immersion, so is f . Using the same argument and 1.7.8, one can show if f 0 is an open immersion, then so is f . Suppose f 0 is a quasi-compact immersion. Then f is quasi-compact by 1.7.3, and f (X) is locally closed in Y by 1.7.9. Let U be an open subset of Y so that f (X) is closed in U , and let U 0 be the inverse image of U in Y 0 . Then f 0 induces a closed immersion X 0 → U 0 . By the above discussion, f induces a closed immersion X → U . So f is an immersion. Proposition 1.8.6. Let g : S 0 → S be a quasi-compact faithfully flat morphism, S 00 = S 0 ×S S 0 , S 000 = S 0 ×S S 0 × S 0 , p1 , p2 : S 00 → S 0 and p12 , p13 , p23 : S 000 → S 00 the projections. Suppose X 0 is an S 0 ∼ = scheme provided with an S 00 -isomorphism σ : p∗1 X 0 → p∗2 X 0 satisfying ∗ ∗ ∗ ∗ 0 p13 (σ) = p23 (σ) ◦ p12 (σ), where pi X (i = 1, 2) are defined by the Cartesian diagrams p∗i X 0 → X 0 ↓ ↓ 00 pi S → S 0.

If X 0 is affine over S 0 , then there exist an S-scheme X and an S 0 ∼ = isomorphism τ : g ∗ X → X 0 such that the diagram p∗ τ

1 p∗1 g ∗ X → p∗1 X 0 ok ↓σ

p∗ τ

2 p∗2 g ∗ X → p∗2 X 0

commutes, where g ∗ X = X ×S S 0 . The S-scheme X is unique up to a unique isomorphism, and X is affine over S. ∼ =

Any S 00 -isomorphism σ : p∗1 X 0 → p∗2 X 0 satisfying p∗13 (σ) = p∗23 (σ) ◦ ∗ p12 (σ) is called a descent datum for the S 0 -scheme X 0 .

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Proof. Let f 0 : X 0 → S 0 be the structure morphism and let A 0 = f∗0 OX 0 . Then A 0 is a quasi-coherent OS 0 -algebra. The descent datum on X 0 defines a descent datum on A 0 . By 1.6.1 (ii), there exist a quasi-coherent OS ∼ = module A and an isomorphism τ : g ∗ A → A 0 such that the diagram p∗ τ

1 p∗1 g ∗ A → p∗1 A 0 ok ↓σ

p∗ τ

2 p∗2 g ∗ A → p∗2 A 0

commutes. By 1.6.1 (i), the morphism A 0 ⊗OS0 A 0 → A 0 defining the multiplication on A 0 can be descended down to a morphism A ⊗OS A → A , and this defines an OS -algebra structure on A . Indeed, the associative law, the commutative law, and the distributive law for A are equivalent to the commutativity of some diagrams involving tensor products of A , and these diagrams commute by 1.6.1 (i) and the commutativity of the corresponding diagrams for A 0 . Set X = Spec A . Then X has the required property. Corollary 1.8.7. Under the assumption of 1.8.4, f is affine if and only if f 0 is affine. Proof. Suppose f 0 is affine. Then f 0 is quasi-compact and separated. By 1.7.3 and 1.7.12, f is quasi-compact and separated. So f∗ OX is a quasicoherent OY -module ([Fu (2006)] 1.4.9 (iii), [EGA] I 9.2.1, [Hartshorne (1977)] II 5.8 (c)). Since the base change is flat, the inverse image of f∗ OX on Y 0 is isomorphic to f∗0 OX 0 ([Fu (2006)] 2.4.10, [EGA] III 1.4.15, [Hartshorne (1977)] III 9.3). The base change of the canonical Y -morphism X → Spec f∗ OX can be identified with the canonical Y 0 -morphism X 0 → Spec f∗0 OX 0 . Since f 0 is affine, the morphism X 0 → Spec f∗0 OX 0 is an isomorphism. By 1.8.4, the morphism X → Spec f∗ OX is an isomorphism. So f is affine. Corollary 1.8.8. Under the assumption of 1.8.4, f is integral, finite, or finite and locally free, if and only if f 0 is so. Proof. We prove the statement about integral morphisms, and leave the rest to the reader. Suppose f 0 is integral. By 1.8.7, f is affine. To prove f is integral, we may reduce to the case where X = Spec B, Y = SpecA, and Y 0 = Spec A0 are affine. We have B = limi Bi , where Bi are finitely −→ generated A-subalgebras of B. Since A0 is flat over A, Bi ⊗A A0 are subalgebras of B ⊗A A0 , and they are finitely generated over A0 . Moreover, we have B ⊗A A0 ∼ B ⊗ A0 . Since B ⊗A A0 is integral over A0 , Bi ⊗A A0 = lim −→i i A

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are finitely generated as A0 -modules. By 1.6.6, Bi are finitely generated as A-modules. So B is integral over A. Hence f is integral. A scheme is called quasi-affine if it is isomorphic to a quasi-compact open subscheme of an affine scheme. A morphism f : X → Y is called quasi-affine if there exists an affine open covering {Vi } of Y such that f −1 (Vi ) are quasi-affine. Any quasi-affine morphism is quasi-compact and separated. Proposition 1.8.9. A morphism f : X → Y is quasi-affine if and only if it is quasi-compact, separated, and the canonical Y -morphism X → Spec f∗ OX is an open immersion. Proof. Note that if f is quasi-compact and separated, then f∗ OX is a quasi-coherent OY -algebra, and we can talk about the Y -scheme Spec f∗ OX . Suppose f is quasi-affine, and let us prove that the Y morphism X → Spec f∗ OX is an open immersion. We may reduce to the case where Y is affine, and prove that if X is a quasi-affine scheme, then the canonical morphism X → Spec Γ(X, OX ) is an open immersion. Suppose X is a quasi-compact open subscheme of an affine scheme Spec A. We can find a1 , . . . , an ∈ A such that X = D(a1 ) ∪ · · · ∪ D(an ). We have a commutative diagram X ,→ Spec A, ↓ % Spec Γ(X, OX ) where the morphism Spec Γ(X, OX ) → Spec A is the morphism induced by the homomorphism A → Γ(X, OX ). For each i, let a0i ∈ Γ(X, OX ) be the image of ai , and let Xa0i = {x ∈ X|a0i is a unit in OX,x }. Restricted to D(ai ), the above diagram induces a commutative diagram Xa0i = D(ai ). ↓ % Spec Γ(X, OX )a0i So Xa0i = D(ai ) is affine, and hence we have Xa0i ∼ = Spec Γ(Xa0i , OX ). On the other hand, we have Γ(Xa0i , OX ) ∼ = Γ(X, OX )a0i . ([Fu (2006)] 1.3.9 (iii), [EGA] I 9.3.3, [Hartshorne (1977)] II 5.14.) It follows that Xa0i → Spec Γ(X, OX )a0i is an isomorphism. Hence X → Spec Γ(X, OX ) is an open immersion.

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Corollary 1.8.10. Under the assumption of 1.8.4, f is quasi-affine if and only if f 0 is so. Proof. Suppose f 0 is quasi-affine. Then f 0 is quasi-compact and separated. This implies that f is quasi-compact and separated. The base change of the canonical Y -morphism X → Spec f∗ OX can be identified with the canonical Y 0 -morphism X 0 → Spec f∗0 OX , which is an open immersion. By 1.8.5, X → Spec f∗ OX is an open immersion. So f is quasi-affine. Proposition 1.8.11. The same as 1.8.6, except that we assume X 0 is quasi-affine over S 0 . Proof. Using 1.6.1, one can show there exists a quasi-coherent OS algebra A such that g ∗ A ∼ = f∗0 OX 0 . By 1.7.8, there exists an open immersion X ,→ Spec A whose base change is the open immersion X 0 ,→ Spec f∗0 OX 0 . Then X has the required property. 1.9

Quasi-finite Morphisms

([SGA 1] I 2, [EGA] II 6.2.) Let (A, m) → (B, n) be a local homomorphism of local rings. We say that B is quasi-finite over A if B/mB is finite dimensional over A/m. Lemma 1.9.1. Let f : X → Y be a morphism of finite type and let x ∈ X. The following conditions are equivalent: (i) x is isolated in f −1 (f (x)). (ii) OX,x is quasi-finite over OY,f (x) . If these conditions hold, we say f is quasi-finite at x. Proof. Replacing the morphism X → Y by f −1 (f (x)) → Spec k(f (x)) does not change OX,x /mf (x) OY,f (x) and OY,f (x) /mf (x)OY,f (x) . So we may assume Y = Spec k for some field k. (i)⇒(ii) Replacing X by a neighborhood of x, we may assume that X consists of one point. Since f is of finite type, we have X = Spec B for some finitely generated k-algebra B with a single prime ideal n. It follows that B/n is finite dimensional over k, and n is nilpotent. Suppose nn = 0. Since B is a noetherian ring, for each 1 ≤ i ≤ n, ni−1 /ni is finite dimensional over B/n, and hence finite dimensional over k. It follows that B is finite dimensional over k.

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(ii)⇒(i) Replacing X by an affine open neighborhood of x, we may assume X = Spec B for some finitely generated k-algebra B. Let q be the prime ideal of B corresponding to the point x. Then Bq is finite dimensional over k. This implies that Bq is an artinian ring, and hence q is a minimal prime ideal of B. Thus x is the generic point of an irreducible component of X. Moreover, Bq /qBq is finite dimensional over k. Hence B/q is finite dimensional over k. So B/q is a field, and q is a maximal ideal of B. Thus x is a closed point of X, and the irreducible component with generic point x consists of only one point. Therefore x is isolated in X. Lemma 1.9.2. Let k be a field and let A be a finitely generated k-algebra. The following conditions are equivalent: (i) Spec A is discrete as a topological space. (ii) Spec A is finite as a set. (iii) A has only finitely many maximal ideals. (iv) A is a finite dimensional k-algebra. Proof. (i)⇒(ii) Spec A is quasi-compact. If it is discrete, it must be finite. (ii)⇒(iii) Trivial. (iii)⇒(i) Let m1 , . . . , mn be all the maximal ideals of A. Since A is a Jacobson ring, any prime ideal of A is the intersection of some maximal Q ideals. Suppose p = ∩j mij is a prime ideal. Then we have p ⊃ j mij . Hence p ⊃ mij for some j. By the maximality of mij , we have p = mij . So any prime ideal of A is a maximal ideal. Hence any point in Spec A is a closed point. As Spec A is finite, it must be discrete. (i)⇒(iv) We have shown that Spec A is a finite discrete topological space. Let m1 , . . . , mn be all the maximal ideals of A. We then have a a Spec A ∼ ··· Spec Amn . = Spec Am1 Hence

A∼ = Am1 × · · · × Amn . By the proof of 1.9.1, each Ami is finite dimensional over k. So A is finite dimensional over k. (iv)⇒(ii) If A is finite dimensional over k, then it is an artinian ring. Hence Spec A is finite. Proposition 1.9.3. Let f : X → Y be a morphism of finite type. The following conditions are equivalent:

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(i) For any y ∈ Y , f −1 (y) is discrete as a topological space. (ii) For any x ∈ X, OX,x is quasi-finite over OY,f (x) . (iii) For any y ∈ Y , f −1 (y) is a finite set. If these conditions hold, we say that f is quasi-finite. Proof. (i)⇔(ii) follows from 1.9.1. (i)⇔(iii) follows from 1.9.2.

Proposition 1.9.4. Consider a Cartesian diagram g0

X ×Y Y 0 → X f0 ↓ ↓f 0 g Y → Y.

Suppose g is quasi-compact and faithfully flat. Then f is quasi-finite if and only if f 0 is so. Proof.

Apply 1.7.4 and 1.9.2.

Lemma 1.9.5. Let (A, m) be a local noetherian ring, and let M be an c= A-module such that M/mM is finite dimensional over A/m. Then M k k b = lim A/m . limk M/m M is finitely generated over A ←− ←−k (0)

(0)

Proof. Let {x1 , . . . , xk } be a family of generators for the A/m-module (i) (i) M/mM . Choose x1 , . . . , xk ∈ M/mi+1 M by induction on i so that their (i−1) (i−1) images in M/mi M are x1 , . . . , xk , respectively. For each i, consider the homomorphism (A/mi+1 )k → M/mi+1 M,

(i)

(i)

(a1 , . . . , ak ) 7→ a1 x1 + · · · + ak xk .

It is surjective by Nakayama’s lemma. Let Ri be its kernel. We have an exact sequence 0 → Ri → (A/mi+1 )k → M/mi+1 M → 0.

Each Ri is an (A/mi+1 )-module of finite length since A/mi+1 is artinian and Ri is finitely generated. By [Fu (2006)] 1.5.1 or [EGA] 0 13.2.2, the sequence 0 → lim Ri → lim(A/mi+1 )k → lim M/mi+1 M → 0 ←− ←− ←− i

i

i

bk → M c. So M c is a is exact. In particular, we have an epimorphism A b finitely generated A-module.

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Lemma 1.9.6. Let (A, m) → (B, n) be a local homomorphism of local b = lim A/mk , and let B b = lim B/nk . noetherian rings, let A ←−k ←−k (i) B is quasi-finite over A if and only if B/n is finite dimensional over A/m, and there exists a positive integer n such that nn ⊂ mB ⊂ n. b is finite over A. b (ii) If B is quasi-finite over A, then B

Proof. (i) Suppose B/mB is finite dimensional over A/m. Then B/n is finite dimensional over A/m. Moreover B/mB is an artinian ring. So its maximal ideal n/mB is nilpotent. Hence there exists a natural number n such that nn ⊂ mB ⊂ n. Conversely, suppose B/n is finite dimensional over A/m, and there exists a natural number n such that nn ⊂ mB ⊂ n. To prove that B/mB is finite dimensional over A/m, it suffices to prove that B/nn is an A-module of finite length. We are then reduced to proving that ni−1 /ni is an A-module of finite length for each 1 ≤ i ≤ n. Since B is noetherian, each ni−1 /ni is finite dimensional over B/n, and hence finite dimensional over A/m. Our assertion follows. (ii) By (i), we have b = lim B/nk ∼ B B/mk B. = lim ←− ←− k

We then apply 1.9.5.

k

Proposition 1.9.7. Let A be a complete noetherian local ring, Y = Spec A, f : X → Y a separated quasi-finite morphism, and x ∈ X a point over the closed point of Y . Then OX,x is finite over A, and the canonical morphism Spec OX,x → X is an open and closed immersion.

Proof. By 1.9.6, ObX,x is finite over A. So OX,x is finite over A. Let g be the canonical morphism Spec OX,x → X. Then f g is finite. Since f is separated, g is finite. So g(Spec OX,x ) is closed in X. On the other hand, g induces an isomorphism of the local ring of X at x with the local ring of Spec OX,x at its closed point. By [Fu (2006)] 1.3.13 (iii) or [EGA] I 6.5.4 (ii), g is an open immersion when restricted to a sufficiently small open neighborhood of the closed point of Spec OX,x . But Spec OX,x is the only open neighborhood of the closed point. Hence g is an open and closed immersion. Corollary 1.9.8. Let A be a complete noetherian local ring, Y = Spec A, and f : X → Y a separated quasi-finite morphism. Then X is the disjoint union of two open and closed subsets X 0 and X 00 such that X 0 is finite over Y , and that the fiber of X 00 → Y over the closed point of Y is empty.

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([EGA] IV 8, [SGA 1] VIII 6.) Throughout this section, we fix a scheme S0 , a direct set I, and a direct system (Aλ , φλµ ), where Aλ (λ ∈ I) are quasi-coherent OS0 -algebras, and φλµ : Aλ → Aµ (λ ≤ µ) are morphisms of OS0 -algebras. Let A = limλ Aλ −→ and let φλ : Aλ → A be the canonical morphisms. Set Sλ = Spec Aλ and S = Spec A . They are S0 -schemes. Let uλµ : Sµ → Sλ be the S0 morphisms induced by φλµ and uλ : S → Sλ the S0 -morphisms induced by φλ . We denote an object over S0 by a symbol with subscript 0, and denote the corresponding object over Sλ (resp. S) induced by base change by the same symbol with subscript λ (resp. without subscript). For example, for any OS0 -module F0 , let Fλ (resp. F ) be the inverse images of F0 on Sλ (resp. S). For any morphism f0 : F0 → G0 of OS0 -modules, let fλ : Fλ → Gλ and f : F → G be the morphisms induced by f0 . For any S0 -scheme X0 , let Xλ = X0 ×S0 Sλ and let X = X0 ×S0 S. For any S0 -morphism f0 : X0 → Y0 of S0 -schemes, let fλ : Xλ → Yλ and f : X → Y be the morphisms induced by f0 . In this section, we prove that given an object over S0 , if its base change to S has some property, then its base changes to Sλ have the same property for sufficiently large λ. We often apply results of this section to the following two situations: (a) Let A be a ring and let {Aλ } be the direct system of subalgebras of A finitely generated by Z. We have A = limλ Aλ . Using results of this section, −→ we can often reduce problems over a general base scheme S = Spec A to problems over a noetherian base schemes Sλ = Spec Aλ . (b) Let S0 be an affine scheme, let x be a point in S0 , let {Sλ } be the inverse system of affine open neighborhoods of x in S0 , and let S = Spec OS0 ,x . We have OS0 ,x = limλ Γ(Sλ , OSλ ). Using results in this section, −→ we can often prove that if the base change to Spec OS0 ,x of an object over S0 has some property, then its base change to a neighborhood of x has the same property. Proposition 1.10.1. (i) S is the inverse limit of the inverse system (Sλ , uλµ ) in the category of schemes. (ii) For any quasi-compact open subset U of S, there exists a quasicompact open subset Uλ of Sλ for some λ such that u−1 λ (Uλ ) = U . (iii) The underlying topological space of S is the inverse limit of the

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inverse system (Sλ , uλµ ) in the category of topological spaces. (iv) If S0 is quasi-compact and S = Ø, then Sλ = Ø for sufficiently large λ. (v) Suppose S0 , Sλ , S are noetherian schemes. Let E0 be a constructible subset of S0 and let Eλ (resp. E) be the inverse images of E0 in Sλ (resp. S). If E = Ø (resp. E = S), then Eλ = Ø (resp. Eλ = Sλ ) for sufficiently large λ. Proof. (i) First we show S is the inverse limit of (Sλ , uλµ ) in the category of S0 schemes. Let f : T → S0 be an S0 -scheme. We need to show the canonical map HomS0 (T, S) → lim HomS0 (T, Sλ ) ←− λ

is bijective. We have HomS0 (T, Sλ ) ∼ = HomOS0 (Aλ , f∗ OT ), ∼ HomS0 (T, S) = HomOS0 (A , f∗ OT ). Since A = limλ Aλ , the canonical map −→ HomOS0 (A , f∗ OT ) → lim HomOS0 (Aλ , f∗ OT ) ←− λ

is bijective. Our assertion follows. Next we show that S is the inverse limit of (Sλ , uλµ ) in the category of schemes. Let T be a scheme. We need to show that the canonical map Hom(T, S) → lim Hom(T, Sλ ) ←− λ

is bijective. Any f ∈ Hom(T, S0 ) defines an S0 -scheme structure on T . Let Homf (T, Sλ ) (resp. Homf (T, S)) be the set of S0 -morphisms with respect to the S0 -scheme structure on T defined by f . We have [ Homf (T, Sλ ), Hom(T, Sλ ) = f ∈Hom(T,S0 )

Hom(T, S) =

[

Homf (T, S).

f ∈Hom(T,S0 )

We have shown that the canonical map Homf (T, S) → lim Homf (T, Sλ ) ←− λ

is bijective. Our assertion follows.

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(ii) It suffices to show that there exists a base of topology for S consisting of open subsets of the form u−1 λ (Uλ ), where Uλ are quasi-compact open subsets of Sλ . We may reduce to the case where S0 is affine. Let Aλ = Γ(S0 , Aλ ) and let A = Γ(S0 , A ). Then Sλ ∼ = Spec Aλ and S ∼ = Spec A. Open subsets of Spec A of the form D(f ) (f ∈ A) form a base of topology for Spec A. Any f ∈ A is the image of some fλ ∈ Aλ for some λ. Then D(f ) is the inverse image of the open subset D(fλ ) of Spec Aλ . Our assertion follows. (iii) The morphisms uλ : S → Sλ induce a map S → limλ Sλ on the ←− underlying topological spaces. To prove that it is a homeomorphism, we may reduce to the case where S0 is affine. Keep the notation in the proof of (ii). The canonical map Spec A → lim Spec Aλ ←− λ

is bijective on the underlying sets. Indeed, given any system of prime ideals pλ of Aλ with φ−1 limλ pλ . Then p is the unique λµ (pµ ) = pλ (λ ≤ µ), let p = − → prime ideal of A with the property φ−1 (p) = pλ . We conclude from (ii) λ that Spec A → limλ Spec Aλ is a homeomorphism. ←− (iv) We may reduce to the case where S0 is affine. Keep the above notation. Since S = Ø, we have 0 = 1 in A. Then we have 0 = 1 in Aλ for sufficiently large λ. So Sλ = Ø for sufficiently large λ. (v) Write E0 = ∪ni=1 (Ui ∩ Fi ), where Ui (resp. Fi ) are open (resp. closed) subsets of S0 . Put the reduced subscheme structures on Ui ∩ Fi . ` Let X0 = ni=1 (Ui ∩ Fi ) and let f0 : X0 → S0 be the canonical morphism. We have E0 = f (X0 ) and hence Eλ = fλ (Xλ ) and E = f (X). If E = Ø, then X = Ø. With (iv) applied to the inverse system (Xλ ), we get Xλ = Ø for sufficiently large λ. Then Eλ = Ø for sufficiently large λ. The second part of (v) follows from the first part applied to the constructible subset S0 − E0 . Proposition 1.10.2. Assume S0 is quasi-compact and quasi-separated. (i) Let F0 and G0 be quasi-coherent OS0 -modules. Suppose F0 locally has finite presentation. Then the canonical map lim HomOSλ (Fλ , Gλ ) → HomOS (F , G ) −→ λ

is bijective. (ii) Suppose F0 and G0 are quasi-coherent OS0 -modules locally of fi∼ = nite presentation. If there exists an isomorphism f : F → G , then for a ∼ = sufficiently large λ, there exists an isomorphism fλ : Fλ → Gλ inducing f .

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(iii) For any quasi-coherent OS -module F locally with finite presentation, there exists a quasi-coherent OSλ -module Fλ locally with finite presentation for a sufficiently large λ such that F ∼ = u∗λ Fλ . We leave the proof to the reader. (Confer the proof of 1.10.9.) Applying 1.10.2 (i) to the case where F = OS0 , we get the following: Corollary 1.10.3. Assume S0 is quasi-compact and quasi-separated. For any quasi-coherent OS0 -module G0 , the canonical map lim Γ(Sλ , Gλ ) → Γ(S, G ) −→ λ

is bijective. Let A be a ring and let B be an A-algebra. We say that B is an Aalgebra of finite presentation if it is isomorphic to A[t1 , . . . , tn ]/I for some nonnegative integer n and some finitely generated ideal I of A[t1 , . . . , tn ]. Note that for any s ∈ A, As ∼ = A[t]/(st − 1) is an A-algebra of finite presentation. Lemma 1.10.4. Let A0 be a ring, (Aλ , φλµ ) a direct system of A0 -algebras, and A = limλ Aλ . −→ (i) Let B0 be an A0 -algebra, Bλ = B0 ⊗A0 Aλ , B = B0 ⊗A0 A, (Cλ ) a direct system of A0 -algebras such that we have a morphism of direct systems from (Aλ ) to (Cλ ), and C = limλ Cλ . If B0 is an A0 -algebra of finite type −→ (resp. finite presentation), then the canonical map lim HomAλ (Bλ , Cλ ) → HomA (B, C) −→ λ

is injective (resp. bijective). (ii) For any A-algebra B with finite presentation, there exists an Aλ algebra Bλ with finite presentation for a sufficiently large λ such that B ∼ = Bλ ⊗Aλ A. Proof. (i) We have one-to-one correspondences HomAλ (Bλ , Cλ ) ∼ = HomA0 (B0 , Cλ ),

HomA (B, C) ∼ = HomA0 (B0 , C).

It suffices to show that lim HomA0 (B0 , Cλ ) → HomA0 (B0 , C) −→ λ

is injective (resp. bijective).

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Suppose B0 is of finite type over A0 , and let x1 , . . . , xn be a finite family of generators of B0 over A0 . If α1 , α2 : B0 → Cλ are A0 -homomorphisms inducing the same homomorphism B0 → C, then α1 (xi ) and α2 (xi ) have the same image in C for each i. Since C = limµ≥λ Cµ , there exists µ ≥ λ −→ such that α1 (xi ) and α2 (xi ) have the same image in Cµ for each i. But then α1 and α2 induce the same homomorphism B0 → Cµ . This proves that our map is injective. Suppose B0 is of finite presentation over A0 . Then there exists an epimorphism π : A0 [t1 , . . . , tn ] → B0 of A0 -algebras such that ker π is a finitely generated ideal of A0 [t1 , . . . , tn ]. Let fj (t1 , . . . , tn ) (j = 1, . . . , m) be a finite family of generators of ker π. Given an A0 -homomorphism α : B0 → C, we have fj (απ(t1 ), . . . , απ(tn )) = απ(fj (t1 , . . . , tn )) = 0. Since C = limλ Cλ , we can find yλ1 , . . . , yλn ∈ Cλ for some λ such that −→ απ(ti ) is the image of yλi in C for each i. Choosing λ sufficiently large, we may assume fj (yλ1 , . . . , yλn ) = 0 (j = 1, . . . , m). Define an A0 homomorphism A0 [t1 , . . . , tn ] → Cλ by mapping ti to yλi for each i. Then ker π is contained in the kernel of this homomorphism. So it induces an A0 -homomorphism αλ : B → Cλ . Its composite with Cλ → C is α. This proves that our map is surjective. (ii) We have B ∼ = A[t1 , . . . , tn ]/I for some n and some finitely generated ideal I of A[t1 , . . . , tn ]. Let fj (t1 , . . . , tn ) (j = 1, . . . , m) be a finite family of generators of I. As A = limλ Aλ , we can find a sufficiently −→ large λ and fλj (t1 , . . . , tn ) ∈ Aλ [t1 , . . . , tn ] whose images in A[t1 , . . . , tn ] are fj (t1 , . . . , tn ), respectively. Let Bλ = Aλ [t1 , . . . , tn ]/(fλ1 , . . . , fλm ). Then B∼ = Bλ ⊗Aλ A. A morphism f : X → Y of schemes is called locally of finite presentation if for any x ∈ X, there exists an affine open neighborhood V = Spec B of f (x) in Y , and an affine open neighborhood U = Spec C of x in f −1 (V ) such that C is a B-algebra with finite presentation. We leave the proof of the following proposition to the reader. Proposition 1.10.5. (i) Local isomorphisms are locally of finite presentation. (ii) If two morphisms f : X → Y and g : Y → Z are locally of finite presentation, then so is gf .

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(iii) If f : X → Y is a morphism locally of finite presentation, then for any morphism g : Y 0 → Y , the base change f 0 : X ×Y Y 0 → Y 0 of f is locally of finite presentation. (iv) Given two morphisms f : X → Y and g : Y → Z, suppose g is locally of finite type and gf is locally of finite presentation. Then f is locally of finite presentation. Proof. (iv) follows from (ii), (iii), and the fact that if g is locally of finite type, then the diagonal morphism ∆ : Y → Y ×Z Y is locally of finite presentation. Indeed, if B is a finitely generated A-algebra, and {b1 , . . . , bn } ⊂ B is a finite family of generators, then the kernel of the epimorphism B ⊗A B → B,

b ⊗ b0 7→ bb0

is generated by 1 ⊗ bi − bi ⊗ 1 (i = 1, . . . , n) as an ideal of B ⊗A B.

Lemma 1.10.6. Let C be a ring and I an ideal of C. Suppose C/I is a C-algebra of finite presentation. Then I is a finitely generated ideal of C. Proof. Let u : C[t1 , . . . , tn ] → C/I be an epimorphism of C-algebras so that ker u is a finitely generated ideal of C[t1 , . . . , tn ]. Choose ci ∈ C (i = 1, . . . , n) such that u(ti ) are the images of ci in C/I respectively. Let v : C[t1 , . . . , tn ] → C be the C-homomorphism defined by v(ti ) = ci and let p : C → C/I be the projection. We have u = pv, and hence ker u = v −1 (I). It is obvious that v is surjective. So we have I = v(v −1 (I)) = v(ker u). Since ker u is a finitely generated ideal of C[t1 , . . . , tn ], I is a finitely generated ideal of C. Corollary 1.10.7. Let A be a ring and B an A-algebra. If Spec B → Spec A is locally of finite presentation, then B is an A-algebra of finite presentation. Moreover, for any epimorphism u : A[t1 , . . . , tn ] → B, ker u is a finitely generated ideal of A[t1 , . . . , tn ]. Proof. Note that Spec B → Spec A is locally of finite type. This implies that B is an A-algebra of finite type. So we have an epimorphism A[t1 , . . . , tn ] → B. Let I be the kernel of this epimorphism. By 1.10.5, the closed immersion Spec A[t1 , . . . , tn ]/I → Spec A[t1 , . . . , tn ]

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is locally of finite presentation. This implies that there exist si ∈ A[t1 , . . . , tn ] (i = 1, . . . , k) such that D(si ) form an open covering of Spec A[t1 , . . . , tn ] and each A[t1 , . . . , tn ]si /Isi is an A[t1 , . . . , tn ]si -algebra of finite presentation. By Lemma 1.10.6, Isi is a finitely generated ideal of A[t1 , . . . , tn ]si . This implies that I is a finitely generated ideal of A[t1 , . . . , tn ]. So B is an A-algebra of finite presentation. Proposition 1.10.8. Let A be a ring and B a finite A-algebra. Then B is an A-algebra of finite presentation if and only if it is an A-module of finite presentation. Proof. Let xi (i = 1, . . . , n) be a finite family of generators of B as an A-module. For each xi , let fi (t) ∈ A[t] be a monic polynomial such that fi (xi ) = 0. Then we have an epimorphism of A-algebras u : A[t1 , . . . , tn ]/(f1 (t1 ), . . . , fn (tn )) → B,

ti 7→ xi .

Note that A[t1 , . . . , tn ]/(f1 (t1 ), . . . , fn (tn )) ∼ = A[t1 ]/(f1 (t1 )) ⊗A · · · ⊗A A[tn ]/(fn (tn )). Hence A[t1 , . . . , tn ]/(f1 (t1 ), . . . , fn (tn )) is a free A-module of finite rank. If B is an A-module of finite presentation, then ker u is a finitely generated A-module, and hence a finitely generated ideal. It follows that B is an A-algebra of finite presentation. Conversely, if B is an A-algebra of finite presentation, then by 1.10.7, ker u is a finitely generated ideal. But A[t1 , . . . , tn ]/(f1 (t1 ), . . . , fn (tn )) is a finitely generated A-module. So ker u is a finitely generated A-module. Hence B is an A-module of finite presentation. A morphism f : X → Y is said to be of finite presentation if it is quasi-compact, quasi-separated, and locally of finite presentation. Proposition 1.10.9. Assume S0 is quasi-compact and quasi-separated. (i) Let X0 and Y0 be S0 -schemes such that X0 → S0 is quasi-compact and quasi-separated, and that Y0 → S0 is locally of finite presentation. Then the canonical map lim HomSλ (Xλ , Yλ ) → HomS (X, Y ) −→ λ

is bijective. (ii) Suppose X0 and Y0 are S0 -schemes of finite presentation. If there ∼ = exists an S-isomorphism f : X → Y , then for a sufficiently large λ, there ∼ = exists an Sλ -isomorphism fλ : Xλ → Yλ inducing f .

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(iii) For any S-scheme X of finite presentation, there exists an Sλ scheme Xλ of finite presentation for a sufficiently large λ such that X ∼ = Xλ ×Sλ S. Proof. (i) Identifying a morphism with its graph, we get one-to-one correspondences HomS (Xλ , Yλ ) ∼ = HomX (Xλ , Xλ ×S Yλ ), λ

λ

λ

HomS (X, Y ) ∼ = HomX (X, X ×S Y ).

It suffices to show that

lim HomXλ (Xλ , Xλ ×Sλ Yλ ) → HomX (X, X ×S Y ) −→ λ

is bijective. We are thus reduced to the case where X0 = S0 , to prove that lim HomSλ (Sλ , Yλ ) → HomS (S, Y ) −→ λ

is bijective. To prove that the above map is injective, we may assume S0 is affine since the problem is local. Let fλ , fλ0 : Sλ → Yλ be two Sλ -morphisms inducing the same S-morphism S → Y by base change. Cover fλ (Sλ ) ∪ (i) (i) fλ0 (Sλ ) by affine open subsets Vλ (i ∈ I) in Yλ and cover each fλ−1 (Vλ ) ∩ (i) (ij) fλ0−1 (Vλ ) by affine open subsets Uλ (j ∈ Ji ) in Sλ . Then U (ij) (i ∈ (ij) I, j ∈ Ji ) form an open covering of S. By 1.10.1 (iv), Uµ (i ∈ I, j ∈ Ji ) form an open covering of Sµ for some µ ≥ λ. Since Sµ is quasi-compact, this open covering has a finite subcovering. Changing notation, we may (i) assume that there exist finitely many affine open subsets Vµ (i = 1, . . . , n) (i) (i) in Yµ , and affine open subsets Uµ (i = 1, . . . , n) in Sµ , such that Uµ cover Sµ and Uµ(i) ⊂ fµ−1 (Vµ(i) ) ∩ fµ0−1 (Vµ(i) )

for all i. Note that the morphisms fµ |U (i) , fµ0 |U (i) : Uµ → Vµ

(i)

induce the

same morphism U → V for any ν ≥ µ, we have

(i) Vλ

are affine,

µ

(i)

(i)

(i)

µ

by base change. Since Sλ and

HomSν (Uν(i) , Vν(i) ) ∼ = HomΓ(Sν ,OSν ) Γ(Vν(i) , OVν(i) ), Γ(Uν(i) , OUν(i) ) , HomS (U (i) , V (i) ) ∼ = HomΓ(S,OS ) Γ(V (i) , OV (i) ), Γ(U (i) , OU (i) ) .

Moreover, by 1.10.3, we have

Γ(U (i) , OU (i) ) ∼ Γ(Uν(i) , OU (i) ). = lim ν −→ ν≥µ

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So by 1.10.4 (i), the map lim HomSν (Uν(i) , Vν(i) ) → HomS (U (i) , V (i) ) −→

ν≥µ

is injective. Hence there exists some ν ≥ µ such that fν |U (i) = fν0 |U (i) for ν

(i)

ν

all i. Since Uν form an open covering of Sν , we have fν = fν0 . This proves that our map is injective. Next we prove that our map is surjective. Let f : S → Y be an Smorphism. For each s ∈ S, choose an affine open neighborhood V0,s of the image of f (s) in Y0 , and an affine open neighborhood W0,s of the image of s in S0 such that the image of V0,s in S0 is contained in W0,s . Let Vs and Ws be the inverse images of V0,s and W0,s in Y and S, respectively. Cover (i) (i) each f −1 (Vs ) by quasi-compact open subsets Us (i ∈ Is ) in S. Then Us (s ∈ S, i ∈ Is ) form an open covering of S. Since S is quasi-compact, this open covering has a finite subcovering. Changing notation, we may (i) assume that there exist finitely many affine open subsets V0 (i = 1, . . . , n) (i) of Y0 , affine open subsets W0 (i = 1, . . . , n) of S0 , and quasi-compact open subsets U (i) of S such that U (i) cover S, f (U (i) ) ⊂ V (i) , and the images (i) (i) V0 in S0 are respectively contained in W0 . By 1.10.1, we can find quasi(i) (i) (i) compact open subsets Uλ of Wλ for some λ such that U (i) = u−1 λ (Uλ ) (i) and that Uλ form an open covering of Sλ . By 1.10.3, we have Γ(Uµ(i) , OU (i) ). Γ(U (i) , OU (i) ) ∼ = lim −→ µ µ≥λ

(i) (i) (i) (i) By 1.10.4 (i), there exist Wµ -morphisms fµ : Uµ → Vµ for some µ ≥ λ inducing the morphisms f |U (i) : U (i) → V (i) . By the injectivity of our map (i) (j) that we have already shown, we have fµ |U (i) ∩U (j) = fµ |U (i) ∩U (j) for all µ µ µ µ (i) pairs (i, j) if µ is sufficiently large. So we can glue fµ together to get a

morphism fµ : Sµ → Yµ inducing f . This proves that our map is surjective. (ii) Let g : Y → X be the inverse of f . By (i), for a sufficiently large λ, we can find Sλ -morphisms fλ : Xλ → Yλ and gλ : Yλ → Xλ inducing f and g, respectively. We have f g = idY and gf = idX . Choosing λ sufficiently large, we may assume fλ gλ = idYλ and gλ fλ = idXλ . Then fλ is an isomorphism. (iii) Cover S0 by finitely many affine open subsets. They induce a finite affine open covering of S. Cover X by finitely many affine open subsets U (i) (i = 1, . . . , n) so that their images in S are contained in members of the above affine open covering of S. By 1.10.4 (ii), there exist Sλ -schemes (i) (i) Uλ for some λ such that Uλ ×Sλ S are S-isomorphic to U (i) , respectively.

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By 1.10.1 (ii), if λ is sufficiently large, we may find quasi-compact open (ij) (i) (ij) subsets Uλ of Uλ such that Uλ ×Sλ S are S-isomorphic to U (i) ∩ U (j) , (ij) respectively. By (ii), if λ is sufficiently large, we have isomorphisms φλ : (ij) ∼ =

(ji)

(ij)

(ji)

(ij)

(ij)

(ik)

(ji)

(jk)

Uλ → Uλ such that φλ = (φλ )−1 , φλ (Uλ ∩Uλ ) = Uλ ∩Uλ , (jk) (ij) (ik) (ij) (ik) (i) and φλ φλ = φλ on Uλ ∩ Uλ . Gluing Uλ together using these isomorphisms, we get an Sλ -scheme Xλ such that Xλ ×Sλ S is S-isomorphic to X. Proposition 1.10.10. Assume S0 is quasi-compact and quasi-separated. Let f0 : X0 → S0 be a morphism of finite presentation. If f is a morphism of one of the types listed below, then for sufficiently large λ, fλ is of the same type. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi)

Open immersion. Closed immersion. Separated. Finite. Affine Surjective. Radiciel. Immersion. Quasi-affine. Quasi-finite. Proper.

Proof. (i) follows from 1.10.1 (ii). (ii) By 1.10.8, f∗ OX is locally of finite presentation as an OS -module, and f \ : OS → f∗ OX is surjective. By 1.10.2 (iii), there exists a quasicoherent OSλ -module Bλ locally of finite presentation for some λ so that f∗ OX ∼ = u∗λ Bλ . By 1.10.2 (i), the morphism f∗ OX ⊗OS f∗ OX → f∗ OX defining the multiplication on f∗ OX is induced by a morphism Bλ ⊗OSλ Bλ → Bλ if λ is sufficiently large. We may assume that this morphism defines an OSλ algebra structure on Bλ . Indeed, the diagrams expressing the associative law, the commutative law and the distributive law commute by 1.10.2 (i) if λ is sufficiently large. Moreover, we can assume that there exists a morphism OSλ → Bλ of sheaves of rings inducing OS → f∗ OX . Let Cλ be the cokernel

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of OSλ → Bλ . Then u∗λ Cλ is the cokernel of OS → f∗ OX , and hence u∗λ Cλ = 0. By 1.10.2 (ii), we may assume Cλ = 0. Then Spec Bλ → Sλ is a closed immersion and it induces f : X → S. By 1.10.9 (ii), we may identify Spec Bλ → Sλ with fλ : Xλ → Sλ if λ is sufficiently large. So fλ is a closed immersion. (iii) Apply (ii) to the diagonal morphism. (iv) Use the same argument as the proof of (ii). (v) The problem is local. We may assume S0 is affine. Then there exists a closed immersion i : X → AnS = Spec OS [t1 , . . . , tn ]

such that πi = f , where π : AnS → S is the projection. By 1.10.9 (i), there exists an Sλ -morphism iλ : Xλ → AnSλ inducing i for some λ. By (ii), we may assume that iλ is a closed immersion. Then fλ is affine. We prove (vi)–(xi) under the extra condition that S0 , Sλ and S are noetherian. The general case is treated in [EGA] IV 8.10.5 and 9.3.3. (vi) Let Eλ = fλ (Xλ ) and let E = f (X). They are constructible by 1.4.2. We have u−1 λ (Eλ ) = E. If f is surjective, then E = S. By 1.10.1(v), we have Eλ = Sλ for a sufficiently large λ. Then fλ is surjective. (vii) Apply (vi) to the diagonal morphisms and use 1.7.1. (viii) f can be factorized as the composite of a closed immersion and an open immersion. Under our extra noetherian condition, we can assume that these two immersions in the factorization are of finite presentation. We then use 1.10.9, (i) and (ii). (ix) The problem is local. We may assume S0 is affine. Let Aλ = Γ(Sλ , OSλ ),

A = Γ(S, OS ),

B = Γ(X, OX ).

By 1.8.9, the canonical morphism X → Spec B is an open immersion. Choose b1 , . . . , bn ∈ B so that X can be identified with the open subset D(b1 ) ∪ · · · ∪ D(bn ) of Spec B. Since X is of finite type over S, each bin Γ(D(bi ), OX ) ∼ = Bbi is a finitely generated A-algebra. Let { bki1i1 , . . . , kinii } bi

bi

be a finite family of generators for Bbi and let C be the A-subalgebra of B generated by bi , bij (i = 1, . . . , n, j = 1, . . . , ni ). Then Cbi ∼ = Bbi . Consider the canonical morphism g : X → Spec C. The image of g is contained in the open subset D(b1 ) ∪ · · · ∪ D(bn ) of Spec C. Moreover, for each i, the homomorphism Γ(D(bi ), OSpec C ) → Γ(D(bi ), OX ) is an isomorphism. So g : X → Spec C is an open immersion. Since C is a finitely generated A-algebra, there exists a closed immersion from Spec C to some AnS . Composed with g, we get an immersion h : X → AnS which is an S-morphism.

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Using 1.10.9 and (viii), one can show that for a sufficiently large λ, there exists an Sλ -morphism hλ : Xλ → AnSλ inducing h, and hλ is an immersion. Then fλ is quasi-affine. (x) The problem is local. We may assume X0 and S0 are affine. Then by the Zariski Main Theorem ([Fu (2006)] 2.5.14, [EGA] III 4.4.3), there exist a finite morphism f¯ : X → S and an open immersion j : X ,→ X such that f = f¯j. By 1.10.9, we can find an Sλ -scheme X λ and an Sλ -morphism jλ : Xλ → X λ inducing j for some λ. By (i) and (iv), we may assume that X λ is finite over Sλ and jλ is an open immersion. Then fλ is quasi-finite. (xi) By Chow’s lemma ([Fu (2006)] 1.4.18, [EGA] II 5.6.1, [Hartshorne (1977)] Exer. II 4.10), there exists a surjective projective morphism g : X 0 → X such that f g is projective. By 1.10.9, for some λ, we can find an Sλ -scheme Xλ0 of finite presentation, and an Sλ -morphism gλ : Xλ0 → Xλ such that X 0 and g are induced from Xλ0 and gλ by base change. Let X 0 → PnS = Proj OS [t0 , . . . , tn ] be a closed immersion so that its composite with the projection PnS → S coincides with f g. By 1.10.9 and (ii), if we choose λ sufficiently large, then we can find a closed immersion Xλ0 → PnSλ = Proj OSλ [t0 , . . . , tn ] so that its composite with the projection PnS → S coincides with fλ gλ . This implies that fλ gλ is proper. Similarly, we can show that gλ is a projective morphism and hence proper if λ is sufficiently large. On the other hand, we may assume that gλ is surjective by (vi). It follows that fλ is proper if λ is sufficiently large. Lemma 1.10.11. Let S be a noetherian scheme, and let f : X → S be a quasi-affine morphism of finite type. Then there exist a projective morphism f¯ : X → S and an open immersion j : X ,→ X such that f = f¯j. Proof. By 1.8.9, the canonical S-morphism X → Spec f∗ OX is an open immersion. By [Fu (2006)] 2.5.2 or [EGA] I 9.4.9, or [Hartshorne (1977)] Exer. II 5.15, we have f∗ OX = limi Ei , where Ei are coher−→ ent OS -submodule of f∗ OX . Let A (Ei ) be the OS -subalgebra of f∗ OX L∞ k generated by Ei . If S(Ei ) = k=1 Sym (Ei ) is the symmetric product of Ei , then Ai is the image of the canonical morphism S(Ei ) → f∗ OX . In particular, A (Ei ) is quasi-coherent. Using the same argument as the proof of 1.10.10 (ix), one can show that when Ei is sufficiently large, the canonical morphism X → Spec A (Ei ) is an open immersion. We have

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a closed immersion Spec A (Ei ) → Spec S(Ei ). So there exists an immersion X → Spec S(Ei ). Composed with the canonical open immersion Spec S(Ei ) → Proj S(Ei ⊕OS ), we get an immersion X → Proj S(Ei ⊕OS ), which is an S-morphism. Let X be the scheme theoretic image of this morphism. Then X is projective over S, and X is an open subscheme of X. Lemma 1.10.12. Let S be a noetherian scheme, and let f : X → S be a separated quasi-finite morphism. Then f is quasi-affine. Proof. We use induction on dim S. To show that f is quasi-affine, it suffices to show that for any s ∈ S, there exists a neighborhood V of s such that f −1 (V ) → V is quasi-affine. Let {Vi } be the family of affine open neighborhoods of s in S. We have OS,s ∼ Γ(Vi , OS ). If we can show = lim −→i that X ×S Spec OS,s → Spec OS,s is quasi-affine, then by 1.10.10 (ix), there exists some Vi such that f −1 (Vi ) → Vi is quasi-affine. So it suffices to treat the case where S = b → Spec A Spec A for some local noetherian ring A. The morphism Spec A b is the completion of A. By is quasi-compact and faithfully flat, where A 1.8.10, it suffices to show that b → Spec A b X ×S Spec A

is quasi-affine. We are thus reduced to the case where A is a complete ` local noetherian ring. By 1.9.8, we have X = X 0 X 00 such that X 0 → S is finite, and the fiber of X 00 → S above the closed point of S is empty. The image of X 00 → S is contained in an open subset of S of dimension ≤ dim S − 1. By the induction hypothesis, X 00 → S is quasi-affine. So X → S is quasi-affine. Theorem 1.10.13 (Zariski Main Theorem). Let S be a noetherian scheme, and let f : X → S be a separated quasi-finite morphism. Then there exist a finite morphism f¯ : X → S and an open immersion j : X ,→ X such that f = f¯j. Proof. Use 1.10.11, 1.10.12 and the Zariski Main Theorem in [Fu (2006)] 2.5.14 or [EGA] III 4.4.3. The Zariski Main Theorem 1.10.13 still holds if we only assume S is quasi-compact and quasi-separated. Confer [EGA] IV 18.12.13.

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Remark 1.10.14. Let (Sλ , uλµ )λ∈I be an inverse system in the category of schemes such that uλµ : Sµ → Sλ (λ ≤ µ) are affine. Then the inverse limit S of this system exists, and all the results in this section hold for such a system. Indeed, fix λ0 ∈ I and take S0 = Sλ0 . We can apply the results of this section to the inverse system (Sλ , uλµ )λ≥λ0 and we have limλ∈I Sλ = limλ∈I Sλ . ←− ←−

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Etale Morphisms and Smooth Morphisms

2.1

The Sheaf of Relative Differentials

Let A → B be a homomorphism of rings, and let M be a B-module. An A-derivation is a map D : B → M satisfying D(b1 + b2 ) = D(b1 ) + D(b2 ), D(b1 b2 ) = b1 D(b2 ) + b2 D(b1 ), D(a) = 0 for any a ∈ A and b1 , b2 ∈ B. The module of relative differentials of B over A is a B-module ΩB/A together with an A-derivation d : B → ΩB/A such that for any B-module M and any A-derivation D : B → M , there exists a unique B-module homomorphism φ : ΩB/A → M such that D = φ ◦ d. Let DerA (B, M ) be the set of A-derivations from B to M . The functor M 7→ DerA (B, M ) is represented by ΩB/A . Recall that a covariant (resp. contravariant) functor F : C → (Sets) from a category C to the category of sets is called representable if there exists an object X in C such that F is isomorphic to the functor HomC (X, −) (resp. HomC (−, X)). In this case, we say F is represented by X. Let I be the kernel of the epimorphism B ⊗A B → B, b1 ⊗ b2 → b1 b2 . We have three ways to put a B-module structure on I/I 2 . (a) I/I 2 is a (B ⊗A B)/I-module. We have a canonical isomorphism (B ⊗A B)/I ∼ = B. This gives a B-module structure on I/I 2 . (b) By multiplication on the left, B ⊗A B is a B-module, and I and I 2 are submodules. This induces a B-module structure on I/I 2 . (c) By multiplication on the right, B ⊗A B is a B-module, and I and I 2 are submodules. This induces a B-module structure on I/I 2 . One verifies that these three B-module structures are the same. Define d : B → I/I 2 by d(b) = 1 ⊗ b − b ⊗ 1 mod I 2 . 59

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Then d is an A-derivation. Proposition 2.1.1. Notation as above. The pair (I/I 2 , d) is a module of relative differentials of B over A. Proposition 2.1.2. Let A be a ring, let A0 and B be A-algebras, and let B 0 = B ⊗A A0 . We have ∼ ΩB/A ⊗B B 0 . ΩB 0 /A0 =

For any multiplicative subset S of B, we have ∼ S −1 ΩB/A . ΩS −1 B/A =

Proof. Let us prove the second statement. Let φ : B → S −1 B be the canonical homomorphism. For any S −1 B-module N , the canonical map DerA (S −1 B, N ) → DerA (B, N ),

D 7→ D ◦ φ

is bijective. In fact, for any derivation D ∈ DerA (B, N ), set sD(b) − bD(s) 0 b = . D s s2 Then D0 ∈ DerA (S −1 B, N ) and D 7→ D0 defines an inverse of the above map. We have DerA (S −1 B, N ) ∼ = HomS −1 B (ΩS −1 B/A , N ), DerA (B, N ) ∼ = HomB (ΩB/A , N ) ∼ = HomS −1 B (S −1 ΩB/A , N ).

It follows that we have a one-to-one correspondence HomS −1 B (ΩS −1 B/A , N ) ∼ = HomS −1 B (S −1 ΩB/A , N )

which is functorial in N . Hence ΩS −1 B/A ∼ = S −1 ΩB/A .

Proposition 2.1.3. Let A → B → C be homomorphisms of rings. Then we have a natural exact sequence of C-modules ΩB/A ⊗B C → ΩC/A → ΩC/B → 0. Proof. Denote the homomorphism B → C by φ. For any C-module N , the kernel of the homomorphism DerA (C, N ) → DerA (B, N ),

D 7→ D ◦ φ

is exactly DerB (C, N ). So we have an exact sequence

0 → DerB (C, N ) → DerA (C, N ) → DerA (B, N ).

Hence we have an exact sequence

0 → HomC (ΩC/B , N ) → HomC (ΩC/A , N ) → HomC (ΩB/A ⊗B C, N ).

This sequence is functorial in N . Our assertion follows.

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Proposition 2.1.4. Let A → B be a homomorphism of rings, let I be an ideal of B, and let C = B/I. Then we have a natural exact sequence of C-modules δ

I/I 2 → ΩB/A ⊗B C → ΩC/A → 0, where δ is induced by the map I → ΩB/A ⊗B C,

b 7→ d(b) ⊗ 1.

Proof. Let N be a B/I-module. For any derivation D : B → N , we have D|I 2 = 0 and D induces a B/I-module homomorphism I/I 2 → N . We thus have an exact sequence 0 → DerA (B/I, N ) → DerA (B, N ) → HomB/I (I/I 2 , N ). So we have an exact sequence 0 → HomC (ΩC/A , N ) → HomC (ΩB/A ⊗A C, N ) → HomC (I/I 2 , N ). This sequence is functorial in N . Our assertion follows.

Proposition 2.1.5. Let k be a field, and let (B, n) be a local k-algebra such that the homomorphism k → B induces an isomorphism k ∼ = B/n. Then the canonical map δ : n/n2 → ΩB/k ⊗B B/n is an isomorphism. Proof. Since k ∼ = B/n, the map δ is surjective by 2.1.4. To show it is injective, it suffices to show that δ ∗ : Homk (ΩB/k ⊗B B/n, k) → Homk (n/n2 , k) is surjective, that is, Derk (B, k) → Homk (n/n2 , k)

is surjective. Let φ : n/n2 → k be a k-homomorphism. For any x ∈ B, we can find a ∈ k such that x ≡ a mod n. Define D(x) = φ(x − a). Then D : B → k is a derivation and δ ∗ (D) = φ. Proposition 2.1.6. If B = A[t1 , . . . , tn ] is a polynomial ring over A, then ΩB/A is a free B-module of rank n with basis {dt1 , . . . , dtn }. Suppose A is an algebra over a ring K, then the sequence 0 → ΩA/K ⊗A B → ΩB/K → ΩB/A → 0 is split exact.

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Proof.

The map n

d:B→B ,

d(f ) =

∂f ∂f ,..., ∂t1 ∂tn

is a B-derivation. Let us show that the pair (B n , d) satisfies the universal property for the module of relative differentials. Let M be a B-module, let D : B → M be an A-derivation, and let xi = D(ti ). Then for any f ∈ B, we have n X ∂f D(f ) = xi . ∂t i i=1 The homomorphism

φ : B n → M,

(b1 , . . . , bn ) 7→ b1 x1 + · · · + bn xn

is the unique homomorphism such that D = φ ◦ d. To get the required split short exact sequence, it suffices to show that any K-derivation D0 : A → M can be extended to a K-derivation D : B → M . Indeed, fix xi ∈ M (i = 1, . . . , n). We can define X D ai1 ...,in ti11 · · · tinn i1 ,...,in

=

X

D0 (ai1 ...,in )ti11

i1 ,...,in

for any

P

i1 ,...,in

ai1 ...,in ti11

· · · tinn

n X

∂ + xi ∂t i i=1

X

ai1 ...,in ti11

i1 ,...,in

· · · tinn ∈ B.

· · · tinn

Corollary 2.1.7. Let C be an A-algebra of finite type (resp. of finite presentation), then ΩC/A is a C-module of finite type (resp. of finite presentation). Proof. We may assume C = A[t1 , . . . , tn ]/I for some n and some ideal I of A[t1 , . . . , tn ]. Let B = A[t1 , . . . , tn ]. By 2.1.4, we have an exact sequence δ

I/I 2 → ΩB/A ⊗B C → ΩC/A → 0. By 2.1.6, ΩB/A ⊗B C is a free C-module of finite rank. When C has finite presentation over A, I/I 2 is a finitely generated C-module. Our assertion follows. Proposition 2.1.8. Let L/K be a finite separable extension of fields. Then ΩL/K = 0.

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Proof. Let M be an L-module, and let D : L → M be a K-derivation. For any x ∈ L, let f (t) be the minimal polynomial of x over K. Then we have f (x) = 0 and f 0 (x) 6= 0. On the other hand, we have 0 = D(f (x)) = f 0 (x)D(x). It follows that D(x) = 0. So DerK (L, M ) = 0. This is true for any Lmodule M . So ΩL/K = 0. Let f : X → Y be a morphism of schemes, and let ∆ : X → X ×Y X be the diagonal morphism. There exists an open subset U of X ×Y X containing the image of ∆ such that ∆ induces a closed immersion ∆ : X → U . Let I be the ideal sheaf of this closed immersion. We define the sheaf of relative differentials of X over Y to be the sheaf ΩX/Y = ∆−1 (I /I 2 ). For all affine open subsets W = Spec B of X and V = Spec A of Y such that f (W ) ⊂ V , ΩX/Y |W can be identified with Ω∼ B/A . ΩX/Y is a quasi-coherent OX -module. If f is locally of finite type (resp. locally of finite presentation), then ΩX/Y is an OX -module locally of finite type (resp. locally of finite presentation). Proposition 2.1.9. Consider a Cartesian diagram p1

X 0 = X ×Y Y 0 → X ↓ ↓ Y 0 → Y. We have p∗1 ΩX/Y ∼ = ΩX 0 /Y 0 . Proposition 2.1.10. Let f : X → Y and Y → Z be morphism of schemes. We have an exact sequence of OX -modules f ∗ ΩY /Z → ΩX/Z → ΩX/Y → 0. Proposition 2.1.11. Let X → Y be a morphism and let i : Z → X be a closed immersion with ideal sheaf I . Then we have an exact sequence of OZ -modules i∗ (I /I 2 ) → i∗ ΩX/Y → ΩZ/Y → 0. Proposition 2.1.12. Let S be a scheme, and let AnS = Spec OS [t1 , . . . , tn ]. Then we have ΩAnS /S ∼ = OAnn . S

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Etale Cohomology Theory

Unramified Morphisms

([SGA 1] I 3, [EGA] IV 17.4.) Proposition 2.2.1. Let f : X → Y be a morphism locally of finite type, x a point in X, and y = f (x). The following conditions are equivalent: (i) my OX,x = mx and the residue field k(x) is a finite separable extension of the residue field k(y). (ii) (ΩX/Y )x = 0. (iii) The diagonal morphism ∆ : X → X ×Y X is an open immersion in a neighborhood of x. Proof. (i)⇒(ii) We have OX,x ⊗OY,y k(y) ∼ = OX,x /my OX,x = OX,x /mx = k(x). By 2.1.2 and 2.1.8, we have (ΩX/Y )x ⊗OX,x OX,x /my OX,x ∼ = Ωk(x)/k(y) = 0. Since f is locally of finite type, (ΩX/Y )x is a finitely generated OX,x -module by 2.1.7. We then apply Nakayama’s lemma. (ii)⇒(iii) Let V be an open subset containing im(∆) such that ∆ induces a closed immersion X → V , and let I be the ideal sheaf of this closed immersion. We have (I /I 2 )∆(x) ∼ = (ΩX/Y )x = 0. Using the assumption that f is locally of finite type, one can verify that I∆(x) is a finitely generated OX×Y X,∆(x) -module. (Confer the proof of 1.10.5 (iv).) By Nakayama’s lemma, we have I∆(x) = 0. But then I vanishes in a neighborhood of ∆(x). So ∆ is an isomorphism in a neighborhood of x. (iii)⇒(i) By the base change Spec k(y) → Y , we are reduced to the case where Y = Spec k for some field k. Replacing X by a neighborhood of x, we may assume ∆ is an open immersion. First consider the case where k is algebraically closed. For any closed point t of X, we have a k-morphism t : Spec k → X with image t. Let Γt be its graph. We have a Cartesian diagram Γ

Spec k →t Spec k ×Spec k X ↓ ↓ X

∆

→

X ×Spec k X.

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Since ∆ is an open immersion, Γt is also an open immersion. We have an isomorphism Spec k ×Spec k X ∼ = X, and through this isomorphism, Γt is identified with t : Spec k → X. So t is an open immersion. This shows that every closed point of X is isolated and Γ(t, OX ) = k. This implies that X is a disjoint union of copies of Spec k. In general, let k¯ be an algebraic closure of k. The above argument shows that after shrinking X, X ⊗k k¯ is isomorphic to a disjoint union of ¯ So X is finite as a set. Replacing X by a finitely many copies of Spec k. neighborhood of x, we may assume that X consists of a single point. By 1.9.2, X = Spec A for some finite dimensional k-algebra A. We then have ¯ So A ⊗k k¯ is reduced and hence A is reduced. As A ⊗k k¯ ∼ = k¯ × · · · × k. X consists of a single point, A is necessarily a field that is finite separable over k. Let f : X → Y be a morphism locally of finite presentation and let x ∈ X. If the conditions in 2.2.1 hold, then we say f is unramified at x. The set of points where f is unramified is open. If f is unramified at every point, we say f is unramified. Proposition 2.2.2. In the following, morphisms are assumed to be locally of finite presentation. (i) Unramified morphisms are locally quasi-finite. (ii) Immersions are unramified. (iii) Composites of unramified morphisms are unramified. (iv) Base changes of unramified morphisms are unramified. (v) If the composite X → Y → Z is unramified, then so is X → Y . Proposition 2.2.3. Let S0 be a quasi-compact quasi-separated scheme, (Aλ , φλµ ) a direct system of quasi-coherent OS0 -algebras, A = limλ Aλ , −→ Sλ = Spec Aλ , S = Spec A , X0 an S0 -scheme, Xλ = X0 ×S0 Sλ , and X = X ×S0 S. Assume that X0 is of finite presentation over S0 . If X is unramified over S, then Xλ is unramified over Sλ for some λ. Proof. Let uλ : X → Xλ be the canonical morphism. We have u∗λ (Ω1Xλ /Sλ ) ∼ = ΩX/S . Since X/S is unramified, we have ΩX/S = 0. By 1.10.2 (ii), we have ΩXλ /Sλ = 0 for some λ. Then Xλ is unramified over Sλ . f

Proposition 2.2.4. Let X → Y → Z be two morphisms. If Y → Z if unramified, then the graph Γf : X → X ×Z Y is an open immersion.

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Proof.

We have a Cartesian diagram Γf

X → X ×Z Y ↓ ↓ ∆

Y → Y ×Z Y.

We then apply 2.2.1 (iii).

Proposition 2.2.5. Consider a Cartesian diagram g0

X ×Y Y 0 → X f0 ↓ ↓f 0 g Y → Y. Suppose f : X → Y is locally of finite presentation and g : Y 0 → Y is faithfully flat. If f 0 is unramified, then so is f . Proof.

We have g 0∗ ΩX/Y ∼ = ΩX×Y Y 0 /Y 0 = 0.

Since g 0 is faithfully flat, this implies that ΩX/Y = 0. 2.3

Etale Morphisms

([SGA 1] I 4, [EGA] IV 17.6–9, 18.1–4.) Let f : X → Y be a morphism locally of finite presentation and let x be a point in X. We say f is etale at x if f is flat at x and unramified at x. If f is etale at every point, we say f is etale. Proposition 2.3.1. In the following, morphisms are assumed to be locally of finite presentation. (i) Open immersions are etale. (ii) Composites of etale morphisms are etale. (iii) Base changes of etale morphisms are etale. (iv) Let f : X → Y and g : Y → Z be two morphisms such that gf is etale and g is unramified. Then f is etale. Proof. Let us prove (iv). By 2.2.4, the graph Γf : X → X ×Z Y of f is an open immersion. By (iii), the projection p2 : X ×Z Y → Y is etale. By (ii), f = p2 Γf is etale.

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Remark 2.3.2. The same argument as the proof of 2.3.1 (iv) shows that the following is true: Let f : X → Y and g : Y → Z be two morphisms such that gf is flat at x ∈ X and g is unramified. Then f is flat at x. Proposition 2.3.3. Let A be a ring, F (t) ∈ A[t] a monic polynomial, and B = A[t]/(F (t)). The canonical morphism Spec B → Spec A is etale at a prime ideal q ∈ Spec B if and only if q does not contain the image of F 0 (t) in B. Hence Spec B → Spec A is etale if and only if the image of F 0 (t) in B is a unit, or equivalently, the ideal generated by F (t) and F 0 (t) is A[t]. Proof. Since F (t) is monic, A[t]/(F (t)) is flat over A. By 2.1.4 and 2.1.6, we have ΩB/A ∼ = A[t]/(F (t), F 0 (t)). It follows that (ΩB/A )q = 0 if and only if q does not contain the image of F 0 (t) in B. Lemma 2.3.4. Let (A, m) be a local ring, B 0 a finite A-algebra, n0 a maximal ideal of B 0 , u an element of B 0 not lying in n0 but lying in all other maximal ideals of B 0 distinct from n0 , B = A[u], and n = n0 ∩ B. Assume that Bn0 0 /mBn0 0 is generated by the image of u in Bn0 0 /mBn0 0 as an A/malgebra. Then the canonical homomorphism Bn → Bn0 0 is an isomorphism. Proof. Let S = B − n and S 0 = B 0 − n0 . We need to show that S −1 B → S 0−1 B 0 is an isomorphism. First let us show S −1 B 0 → S 0−1 B 0 is an isomorphism. It suffices to show that every element in S 0 is invertible in S −1 B 0 , or equivalently, the images in S −1 B 0 of elements in S 0 do not lie in any maximal ideal of S −1 B 0 . Since S −1 B 0 is finite over S −1 B, any maximal ideal of S −1 B 0 lies above the maximal ideal of S −1 B = Bn , and hence above the maximal ideal n of B. But B 0 is finite over B. So any maximal ideal of S −1 B 0 lies above a maximal ideal of B 0 , and hence is of the form S −1 n00 for some maximal ideal n00 of B 0 satisfying n00 ∩ B = n. If u ∈ n00 , then u ∈ n00 ∩ B = n ⊂ n0 . This contradicts to our assumption that u 6∈ n0 . So u 6∈ n00 . As u lies in maximal ideals of B 0 distinct from n0 , we must have n0 = n00 . Hence S −1 n0 is the only maximal ideal of S −1 B 0 . It is clear that the images in S −1 B 0 of elements in S 0 do not lie in S −1 n0 . So S −1 B 0 → S 0−1 B 0 is an isomorphism. Since B is a subring of B 0 , S −1 B → S −1 B 0 is injective. Hence S −1 B → S 0−1 B 0 is injective. To prove it is surjective, it suffices to show that Bn /mBn → Bn0 0 /mBn0 0 is surjective by Nakayama’s lemma. This follows from the assumption that the image of u in Bn0 0 /mBn0 0 is a generator over A/m.

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Theorem 2.3.5 (Chevalley). Let f : X → Y be a morphism locally of finite presentation, x a point in X, and A = OY,f (x) . Then f is etale at x if and only if we can find a monic polynomial F (t) ∈ A[t], a maximal ideal n of B = A[t]/(F (t)) not containing the image of F 0 (t) in B such that OX,x is A-isomorphic to Bn . Proof. The “if” part follows from 2.3.3. To prove the “only if” part, we make the base change Spec A → Y to reduce to the case where Y = Spec A. Let m be the maximal ideal of A. First suppose f is unramified at x. Let U be an affine open neighborhood of x in X such that f |U : U → Spec A is unramified. By the Zariski Main Theorem 1.10.13, f |U can be factorized as a composite j

f¯

U ,→ SpecB 0 → SpecA such that j is an open immersion and f¯ is a finite morphism. (We only prove the Zariski Main Theorem 1.10.13 under the noetherian assumption. Write A = limλ Aλ , where Aλ are subalgebras of A finitely generated over −→ Z. By 1.10.4 (ii) and 2.2.3, we can find an unramified affine morphism fλ : Uλ → Spec Aλ of finite type such that f |U is obtained from fλ by base change. Applying 1.10.13 to the morphism fλ and taking base change, we get the above factorization of f |U .) Let n0 be the prime ideal of B 0 corresponding to the image of x in Spec B 0 . Since it is above the maximal ideal m of A, n0 is a maximal ideal of B 0 . Note that OX,x is A-isomorphic to Bn0 0 . Let n01 , . . . , n0r be all the maximal ideals of B 0 distinct from n0 . Since B 0 /n0 is a finite separable extension of A/m, it is generated by a single element. By the Chinese remainder theorem, we can find u ∈ n01 ∩ · · · ∩ n0r such that the image of u in B 0 /n0 is nonzero and generates B 0 /n0 over A/m. Let B = A[u] and let n = n0 ∩ B. By 2.3.4, OX,x is A-isomorphic to Bn . Let u ¯ be the image of u in B/mB, let f (t) ∈ (A/m)[t] be the minimal polynomial of u¯ over A/m, and let d = deg f . We have B/mB ∼ = (A/m)[t]/(f (t)). By Nakayama’s lemma, B is generated by 1, u, . . . , ud−1 as an A-module. So there exists a monic polynomial F (t) ∈ A[t] of degree d such that F (u) = 0. The image of F (t) in (A/m)[t] is necessarily f (t). Since A/m is a field, Spec B/mB → Spec A/m is etale at n/mB. By 2.3.3, we have f 0 (¯ u) 6∈ 0 n/mB. So we have F (u) 6∈ n. Suppose furthermore that f is flat at x. Then Bn is flat over A. Let n0 be the inverse image of n under the epimorphism A[t]/(F (t)) → B,

t 7→ u

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of A-algebras. Then n0 does not contain the image of F 0 (t) in A[t]/(F (t)). By 2.3.3, Spec A[t]/(F (t)) → Spec A is etale at n0 , and in particular unramified in a neighborhood of n0 . Applying 2.3.2 to the composite Spec B → Spec A[t]/(F (t)) → Spec A, we see that the morphism Spec B → Spec A[t]/(F (t)) is flat at n. So (A[t]/(F (t)))n0 → Bn is faithfully flat and hence injective. But A[t]/(F (t)) → B is surjective, so we must have (A[t]/(F (t)))n0 ∼ = Bn . Hence (A[t]/(F (t)))n0 is A-isomorphic to OX,x . Corollary 2.3.6. Let f : X → Y be a morphism locally of finite presentation. The set of points in X where f is etale is open. Proof. Suppose f is etale at x ∈ X. By 2.3.5, we can find a monic polynomial F (t) ∈ OY,f (x) [t] and a maximal ideal of n of B = OY,f (x) [t]/(F (t)) not containing the image of F 0 (t) in B such that OX,x is OY,f (x) -isomorphic to Bn . Let V = Spec A be an affine open neighborhood of f (x) in Y such that F (t) is the image of a monic polynomial in A[t] under the canonical homomorphism A[t] → OY,f (x) [t], and denote such a monic polynomial in A[t] also by F (t). Let p be the inverse image of n under the canonical homomorphism A[t]/(F (t)) → OY,f (x) [t]/(F (t)). Then Spec A[t]/(F (t)) → V is unramified at p, and hence unramified in an open neighborhood W of p in Spec A[t]/(F (t)). As A[t]/(F (t)) is a flat A-algebra, Spec A[t]/(F (t)) → V is etale in W . Note that OX,x is OY,f (x) -isomorphic to (A[t]/(F (t))p . By [Fu (2006)] 1.3.13 (iii) or [EGA] I 6.5.4 (ii), there exists an open neighborhood U of x in X and an open neighborhood W 0 of p in W such that U and W 0 are Y -isomorphic. Then f is etale in U . Proposition 2.3.7. Let S0 be a quasi-compact quasi-separated scheme, (Aλ , φλµ ) a direct system of quasi-coherent OS0 -algebras, A = limλ Aλ , −→ Sλ = Spec Aλ , S = Spec A , X0 an S0 -scheme, Xλ = X0 ×S0 Sλ , and X = X ×S0 S. Assume that X0 is of finite presentation over S0 . If X is etale over S, then Xλ is etale over Sλ for some λ. Proof. Let x be a point in X and let s be its image in S. By 2.3.5, we can find a monic polynomial F (t) ∈ OS,s [t] and a maximal ideal n of B = OS,s [t]/(F (t)) not containing the image of F 0 (t) in B such that OX,x is OS,s -isomorphic to Bn . Let V = Spec A be an affine open neighborhood of s in S such that F (t) is the image of a monic polynomial in A[t] under

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the canonical homomorphism A[t] → OS,s [t], and denote such a monic polynomial in A[t] also by F (t). Let p be the inverse image of n under the canonical homomorphism A[t]/(F (t)) → OS,s [t]/(F (t)). Then OX,x is OS,s -isomorphic to (A[t]/(F (t)))p . We can find a quasi-compact open neighborhood U of x in X whose image in S is contained in V , and a quasicompact open neighborhood W of p in Spec A[t]/(F (t)) such that U and W are V -isomorphic by [Fu (2006)] 1.3.13 (iii) or [EGA] I 6.5.4 (ii). By 1.10.1 (ii), there exists an open subset Vλ of Sλ for some λ whose inverse image in S is V . Shrinking Vλ and V , we may assume Vλ = Spec Aλ is affine. Taking λ sufficiently large, we may assume F (t) is the image in A[t] of a monic polynomial Fλ (t) ∈ Aλ [t] such that the inverse image pλ of p in Aλ [t]/(Fλ (t)) does not contain the image of Fλ0 (t) in Aλ [t]/(Fλ (t)). Again by 1.10.1 (ii), we may assume that U (resp. W ) is the inverse image of an open subset Uλ (resp. Wλ ) of Xλ (resp. Spec Aλ [t]/(Fλ (t))) in X (resp. in Spec A[t]/(F (t))), and by 1.10.9 (ii), we may assume that Uλ is Vλ -isomorphic to Wλ . But Wλ → Vλ is etale at pλ by 2.3.3. So Xλ is etale over Sλ at xλ , where xλ is the image of x in Xλ . For each λ, let Oλ be the largest open subset of Xλ on which Xλ is etale over Sλ , and let uλ : X → Xλ be the projection. The above discussion shows that X = ∪λ u−1 λ (Oλ ). Since −1 X is quasi-compact and u−1 (O ) ⊂ u (O ) for any pair λ ≤ µ, we have λ µ µ λ −1 −1 X = uλ (Oλ ) for some λ, that is, uλ (Xλ − Oλ ) = Ø. By 10.1.1 (iv), there exists µ ≥ λ such that Xµ − Oµ = Ø. Then Xµ is etale over Sµ . Proposition 2.3.8. Etale morphisms are open mappings. Proof. The problem is local. It suffices to prove that any affine etale morphism f : Spec B → Spec A is an open mapping. By 1.10.4 (ii) and 2.3.7, we can find a subalgebra A0 of A finitely generated over Z, and an algebra B0 etale and finitely generated over A0 such that we have an Aisomorphism B ∼ = B0 ⊗A0 A. Let us prove that the morphism Spec (B0 ⊗A0 A) → Spec A is an open mapping. Let {Aλ } be the family of subalgebras of A finitely generated over A0 . Fix the notation by the following diagram: vλ Spec (B0 ⊗A0 A) → Spec (B0 ⊗A0 Aλ ) f ↓ ↓ fλ uλ Spec A → Spec Aλ . Each fλ is etale and hence flat. By 1.5.2, fλ is an open mapping. By 1.10.1 (ii), any quasi-compact open subset U of Spec (B0 ⊗A0 A) is of the form U = vλ−1 (Uλ ) for a large λ and an open subset Uλ of Spec (B0 ⊗A0 Aλ ). Since fλ (Uλ ) is open and f (U ) = f vλ−1 (Uλ ) = u−1 λ fλ (Uλ ),

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f (U ) is also open.

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Proposition 2.3.9. A morphism f : X → Y is an open immersion if and only if it is etale and radiciel. Proof. The “only if” part is clear. Let us prove the “if” part. Since f is injective, it suffices to show that f is locally an open immersion. So we may assume that f : X → Y is induced by a base change Y → Spec A0 from an etale morphism f0 : Spec B0 → Spec A0 , where A0 is a finitely generated Z-algebra and B0 is a finitely generated A0 -algebra. By the Zariski Main Theorem 1.10.13, f0 can be factorized as the composite of an open immersion and a finite morphism. So we have a factorization f = f¯j such that j : X ,→ X is an open immersion and f¯ : X → Y is finite. By 2.3.8, f (X) is open. Replacing Y by f (X) and X by f¯−1 (f (X)), we may assume that f is surjective. Since f is radiciel and etale, it is universally injective and universally open, and hence universally a homeomorphism. It follows that f is proper. Then j is proper. Hence j is an open and closed immersion, and in particular a finite morphism. So f is a finite morphism. Let us prove this implies that f is an isomorphism. We may reduce to the case where Y = Spec A is affine. Then X = Spec B for some finite A-algebra B. Since f is radiciel, for any prime ideal p of A, Bp is local. Since f is also etale, we have Ap /pAp ∼ = Bp /pBp . By Nakayama’s lemma, Ap → Bp is surjective. It is injective since it is faithfully flat. So Ap ∼ = Bp for any prime ideal p of A. This implies that A ∼ = B. Corollary 2.3.10. Let f : X → Y be a morphism. (i) If f is unramified and separated and Y is connected, then there is a one-to-one correspondence between the set of sections of f and the set of connected components of X on which f induces an isomorphism. In particular, a section of f is completely determined by its value at one point of Y . (ii) If f is etale, then there is a one-to-one correspondence between the set of sections of f and the set of open subsets of X on which f is radiciel and surjective. Proof. Suppose that f : X → Y is unramified and s : Y → X is a section of f . We have f s = id. It follows that s must be an immersion and etale. By 2.3.9, s is an open immersion, and hence it induces an isomorphism between Y and the open subset s(Y ) of X. If f is separated and Y is connected, then s is an open and closed immersion and it induces an isomorphism between Y and a connected component of X.

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Lemma 2.3.11. Let S be a scheme, S0 a subscheme of S, and X0 → S0 an etale morphism. For any x ∈ X0 , there exists an open neighborhood U0 of x in X0 such that U0 is S0 -isomorphic to U ×S S0 for some etale S-scheme U. Proof. Let s be the image of x in S0 . By 2.3.5, there exists a monic polynomial F0 (t) ∈ OS0 ,s [t] and a maximal ideal n0 of OS0 ,s [t]/(F0 (t)) such that OX0 ,x is OS0 ,s -isomorphic to (OS0 ,s [t]/(F0 (t)))n0 and the image of F00 (t) in OS0 ,y [t]/(F0 (t)) does not lie in n0 . Let F (t) ∈ OS,s [t] be a monic polynomial whose image under the homomorphism OS,s [t] → OS0 ,s [t] is F0 (t), and let n be the inverse image of n0 under this homomorphism. Then Spec OS,s [t]/(F (t)) → Spec OS,s is etale at n. We have (OS,s [t]/(F (t)))n ⊗OS,s OS0 ,s ∼ = (OS0 ,s [t]/(F0 (t)))n0 . There exists an S-scheme U locally of finite presentation and a point z ∈ U above s such that OU,z is OS,s -isomorphic to (OS,s [t]/(F (t)))n . We may assume that U → S is etale. We have an OS0 ,s -isomorphism OU×S S0 ,z ∼ = OX0 ,x . It extends to an isomorphism between a neighborhood U0 of x in X0 and a neighborhood W0 of z in U ×S S0 . Shrinking U , we may assume W0 = U ×S S0 . Proposition 2.3.12. Let S be a scheme, and let S0 be a closed subscheme of S with the same underlying topological space as S. The functor X 7→ X ×S S0 from the category of etale S-schemes to the category of etale S0 schemes is an equivalence of categories. Proof. Let X and Y be S-schemes. Suppose that Y is etale over S. Let us prove that the canonical map HomS (X, Y ) → HomS0 (X ×S S0 , Y ×S S0 ) is bijective. An S-morphism f : X → Y is completely determined by its graph Γf : X → X ×S Y , and Γf is a section of the projection p : X ×S Y → X. Since S0 is a closed subscheme of S with the same underlying topological space as S, by 2.3.10 (ii), there is a one-to-one correspondence between the set of sections of p and the set of sections of the base change p0 of p by S0 → S. Our assertion follows. Let X0 → S0 be an etale morphism. By 2.3.11, there exists an open covering {Uα0 } of X0 such that each Uα0 is S0 -isomorphic to Uα ×S S0 for some etale S-scheme Uα . Note that each Uα has the same underlying

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topological space as Uα0 . By the discussion above, for each pair α, β, the S-scheme structure on Uα0 ∩ Uβ0 induced from Uα is isomorphic to that induced from Uβ . We can glue {Uα } together to get an S-scheme X so that X ×S S0 is S0 -isomorphic to X0 . 2.4

Smooth Morphisms

([SGA 1] II 1–3, [EGA] IV 17.5.) A morphism f : X → Y is called smooth at a point x in X if there exists an open neighborhood U of x such that f |U can be factorized as a composite of an etale morphism U → AnY and the projection AnY → Y for some integer n. Note that n is unique. It is equal to inf V dim (f −1 f (x) ∩ V ), where V goes over the family of open neighborhoods of x in X. We call inf V dim (f −1 f (x) ∩ V ) the relative dimension of f at x. If the relative dimension of f is the constant n on X, we say f is pure of relative dimension n. If f is smooth at every point, we say that f is smooth. Smooth morphisms are flat. Proposition 2.4.1. (i) A morphism is etale if and only if it is smooth and locally quasi-finite. (ii) Composites of smooth morphisms are smooth. (iii) Base changes of smooth morphisms are smooth. (iv) Let f : X → Y and g : Y → Z be morphisms such that g is unramified and gf is smooth. Then f is smooth. Proposition 2.4.2. Let S be a noetherian scheme, X and Y two Sschemes locally of finite type, f : X → Y an S-morphism, x a point in X, s the image of x in S, Xs = X ⊗OS k(s), Ys = Y ⊗OS k(s), and fs : Xs → Ys the morphism induced by f . (i) f is quasi-finite (resp. unramified) at x if and only if fs is quasi-finite (resp. unramified) at x. (ii) Suppose that X and Y are flat over S. Then f is flat (resp. etale, resp. smooth) at x if and only if fs is flat (resp. etale, resp. smooth) at x. Proof. (i) is clear since the property of a morphism being quasi-finite or unramified depends only on fibers of the morphism. The statement for the flat morphism follows from 1.3.7. The statement for the etale morphism follows from that for the flat morphism and for the unramified morphism. The statement for the smooth morphism follows from 2.4.3 below.

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Lemma 2.4.3. Let f : X → Y be a morphism locally of finite type between noetherian schemes, x a point in X and y = f (x). Then f is smooth at x if and only if f is flat at x and fy : Xy = X ⊗OY k(y) → Spec k(y) is smooth at x. Proof. The “only if” part is clear. Let us prove the “if” part. Shrinking X, we may assume there exists an etale k(y)-morphism Xy → Ank(y) = Spec k(y)[t1 , . . . , tn ]. Let si (i = 1, . . . , n) be the images of ti in Γ(Xy , OXy ). Shrinking X again, we may assume that si lie in the image of the canonical homomorphism Γ(X, OX ) → Γ(Xy , OXy ). Choose a preimage s0i ∈ Γ(X, OX ) of si for each i. Consider the Y -morphism X → AnY = Spec OY [t1 , . . . , tn ] corresponding to the OY -algebra morphism OY [t1 , . . . , tn ] → f∗ OX mapping ti to s0i . By the statement for the etale morphism in 2.4.2 (ii), this morphism is etale at x. So X → Y is smooth at x. Proposition 2.4.4. Let f : X → Y be a morphism locally of finite type between noetherian schemes, and let x be a point in X. Suppose f is smooth at x. Then OX,x is reduced (resp. regular, resp. normal) if and only if OY,f (x) is so. Proof. The problem is local. We may assume there exists an etale Y morphism X → AnY . Let y 0 be the image of x in AnY . Note that OAnY ,y0 is reduced (resp. regular, resp. normal) if and only if OY,f (x) is so. (Confer [Matsumura (1970)] (17.B), (17.I) Theorem 40, (21.D) Theorem 51.) We are thus reduced to the case where f is etale. The statement about regularity follows from [Matsumura (1970)] (21.D) Theorem 51. Recall that a noetherian ring is reduced (resp. normal) if it satisfies (R0 ) and (S1 ) (resp. (R1 ) and (S2 )). (Confer [Matsumura (1970)] (17.I) Theorem 39.) The statements about being reduced and being normal follow from [Matsumura (1970)] (21.C) Corollary 2. Proposition 2.4.5. Let S0 be a quasi-compact quasi-separated scheme, (Aλ , φλµ ) a direct system of quasi-coherent OS0 -algebras, A = limλ Aλ , −→ Sλ = Spec Aλ , S = Spec A , X0 an S0 -scheme of finite presentation, Xλ = X0 ×S0 Sλ , and X = X0 ×S0 S. If X is smooth over S, then Xλ is smooth over Sλ for some λ.

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Proof. The problem is local. We may assume there exists an etale Smorphism X → AnS . By 1.10.9 (i), this S-morphism is induced by an Sλ -morphism Xλ → AnS for some λ. If λ is sufficiently large, Xλ → AnS is etale by 2.3.7. Then Xλ is smooth over Sλ . 2.5

Jacobian Criterion

([SGA 1] II 4, [EGA] IV 17.11–12.) Proposition 2.5.1. Let S be a scheme, X and Y two S-schemes, and f : X → Y an S-morphism locally of finite presentation. Then f is unramified if and only if the canonical morphism f ∗ ΩY /S → ΩX/S is surjective. Proof.

Use the exact sequence f ∗ ΩY /S → ΩX/S → ΩX/Y → 0.

Lemma 2.5.2. Let A → B be a local homomorphism of local noetherian rings, let M be a finitely generated B-module, and let t be an element in the maximal ideal of A which is not a zero divisor in A. Then M is flat over A if and only if the homomorphism t : M → M,

x 7→ tx

is injective and M/tM is flat over A/tA. Proof. If M is flat over A, then M/tM is flat over A/tA. Since t is not a zero divisor, the multiplication by t is injective on A. So t : M → M is injective. Conversely, from the short exact sequence t

0 → A → A → A/tA → 0, we get a long exact sequence t

0 → TorA 1 (M, A/tA) → M → M → M/tM → 0.

If t : M → M is injective, then we have TorA 1 (M, A/tA) = 0. Furthermore, if M/tM is flat over A/tA, then M is flat over A by 1.3.5. Proposition 2.5.3. Let S be a scheme, X and Y two S-schemes, and f : X → Y an S-morphism locally of finite presentation. If f is etale, then the canonical morphism f ∗ ΩY /S → ΩX/S is an isomorphism. The converse holds if X and Y are smooth over S.

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Proof. Suppose f is etale and let us prove f ∗ ΩY /S ∼ = ΩX/S . The problem is local. We may assume that X and Y are separated over S. The diagonal morphisms ∆X/S , ∆Y /S and ∆X/Y are closed immersions. We have a commutative diagram ∆X/Y

q

X → X ×Y X → X ×S X &f ↓p ↓ f ×f Y

∆Y /S

→ Y ×S Y,

where the square is Cartesian and q∆X/Y = ∆X/S . Let I be the OY ×S Y ideal defining the closed immersion ∆Y /S . Since f × f is flat, the OX×S X ideal defining the closed immersion q is (f × f )∗ I . Since f is unramified, ∆X/Y is an open immersion. It follows that ΩX/S ∼ = ∆∗X/Y q ∗ (f × f )∗ I /(f × f )∗ I 2 ∼ = ∆∗X/Y q ∗ (f × f )∗ (I /I 2 ) ∼ = f ∗ ∆∗ (I /I 2 ) ∼ = f ∗ ΩY /S . Y /S

Suppose X and Y are smooth over S and f ∗ ΩY /S ∼ = ΩX/S . Let us prove f is etale. The problem is local. We may reduce to the case where X, Y and S are affine. Using 1.10.2 (ii), 1.10.4 (ii) and 2.4.5, we can then reduce to the case where S is noetherian. By 2.5.1, f is unramified. We need to show that f is flat. By 2.4.2 (ii), it suffices to show fs : Xs → Ys is flat for any s ∈ S. So we may assume S = Spec k for some field k. By base change to an algebraic closure of k, we may assume that k is algebraically closed. Since X and Y are smooth over k, they are regular. Let x be a closed point of X, and let y = f (x). We need to show OX,x is flat over OY,y . By 2.1.5, we have canonical isomorphisms mx /m2x ∼ = ΩOX,x /k ⊗OX,x OX,x /mx , 2 ∼ my /m = ΩO /k ⊗O OY,y /my . y

Y,y

Y,y

As f ∗ ΩY /S ∼ = mx /m2x . So if t1 , . . . , tn ∈ my form = ΩX/S , we have my /m2y ∼ a regular system of parameters of OY,y , then their images in OX,x form a regular system of parameters of OX,x . Since OX,x /(t1 , . . . , tn )OX,x is flat over OY,y /(t1 , . . . , tn ) ∼ = k, by 2.5.2 and induction on n, OX,x is flat over OY,y . Proposition 2.5.4. Let S be a scheme, X and Y two S-schemes, and f : X → Y a smooth S-morphism. (i) ΩX/Y is a locally free OX -module, and its rank is equal to the relative dimension of f .

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(ii) The sequence 0 → f ∗ ΩY /S → ΩX/S → ΩX/Y → 0 is exact. Proof. The problem is local. We may assume that f : X → Y is a composite g

π

X → AnY → Y, where g is etale, and π is the projection. By 2.5.3,we have ΩX/Y ∼ = g ∗ ΩAnY /Y ,

ΩX/S ∼ = g ∗ ΩAnY /S .

The sequence in (ii) can be identified with 0 → g ∗ π ∗ ΩY /S → g ∗ ΩAnY /S → g ∗ ΩAnY /Y → 0. This reduces the proof to the case where f is the morphism π : AnY → Y . We can then apply 2.1.6. Corollary 2.5.5. Let S be a scheme, X and Y two S-schemes locally of finite presentation, and f : X → Y an etale S-morphism. If X is etale over S, then f (X) is etale over S. Proof.

By 2.5.4, we have an exact sequence 0 → f ∗ ΩY /S → ΩX/S → ΩX/Y → 0.

Since X is etale over S, we have ΩX/S = 0. So f ∗ ΩY /S = 0. But f is flat. It follows that ΩY /S |f (X) = 0. Hence f (X) is unramified over S. For any x ∈ X, let y and s be its images in Y and S, respectively. Then OX,x is faithfully flat both over OY,y and over OS,s . This implies that OY,y is flat over OS,s . So f (X) is etale over S. Lemma 2.5.6. Let X be a scheme, F and G quasi-coherent OX -modules, x a point in X, and u : F → G a morphism. Suppose that G is free of finite rank in a neighborhood of x. The following conditions are equivalent: (i) F has finite presentation in a neighborhood of x, ux : Fx → Gx is injective and coker ux is a free OX,x -module. (ii) There exists a neighborhood U of x in X such that u|U induces an isomorphism between F |U and a direct factor of G |U . (iii) F has finite presentation in a neighborhood of x and ux is universally injective, that is, for any OX,x -module M , ux ⊗ id : Fx ⊗OX,x M → Gx ⊗OX,x M is injective.

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(iv) F has finite presentation in a neighborhood of x and ux ⊗ id : Fx ⊗OX,x k(x) → Gx ⊗OX,x k(x) is injective. ∨ ∨ (v) F is free in a neighborhood of x, and the transpose u∨ x : Gx → Fx of ux is surjective, where for any free OX,x -module M of finite rank, we define M ∨ = HomOX,x (M, OX,x ). Proof. (i)⇒(ii) The short exact sequence u

0 → Fx →x Gx → coker ux → 0 is split. So there exists a homomorphism vx : Gx → Fx such that vx ux = id. Since F and G have finite presentation in a neighborhood of x, we can find an open neighborhood U of x in X and a morphism v : G |U → F |U extending vx such that v ◦ (u|U ) = id. Then u|U induces an isomorphism between F |U and a direct factor of G |U . (ii)⇒(iii) and (iii)⇒(iv) are obvious. (iv)⇒(v) Choose s1 , . . . , sm ∈ Fx so that their images in Fx ⊗OX,x k(x) form a basis of Fx ⊗OX,x k(x). By Nakayama’s lemma, s1 , . . . , sm generate Fx as an OX,x -module. Let s01 , . . . , s0m ∈ Gx be the images of s1 , . . . , sm , respectively. The images of s01 , . . . , s0m in Gx ⊗OX,x k(x) form a part of a basis of Gx ⊗OX,x k(x) since Fx ⊗OX,x k(x) → Gx ⊗OX,x k(x) is injective. Choose s0m+1 , . . . , s0n ∈ Gx so that the images of s01 , . . . , s0n in Gx ⊗OX,x k(x) form a basis of Gx ⊗OX,x k(x). By Nakayama’s lemma and the assumption that Gx is free, s01 , . . . , s0n form a basis of Gx . This implies that s1 , . . . , sm are linearly independent. Since s1 , . . . , sm generate Fx , they form a basis of Fx . Hence Fx is a free OX,x -module. Since F is of finite presentation in a neighborhood of x, F is free in a neighborhood of x. The homomorphism (ux ⊗ id)∨ : Hom(Gx ⊗OX,x k(x), k(x)) → Hom(Fx ⊗OX,x k(x), k(x))

is surjective. So the homomorphism

∨ ∨ u∨ x ⊗ id : Gx ⊗OX,x k(x) → Fx ⊗OX,x k(x)

is surjective. By Nakayama’s lemma, u∨ x is surjective. (v)⇒(i) Since u∨ is surjective, and Fx is a free OX,x -module, the map x Hom(Gx , Fx ) → Hom(Fx , Fx ) induced by ux is surjective. So there exists vx : Gx → Fx such that vx ux = id. Thus the sequence u

0 → Fx →x Gx → coker ux → 0

is exact and split. It follows that ux is injective and coker ux is a direct factor of Gx . As Gx is a free OX,x -module, so is coker ux .

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Proposition 2.5.7. Let S be a scheme, X and Y two S-schemes, f : X → Y an S-morphism, x a point in X, and y and s the images of x in Y and S, respectively. Suppose that Y is smooth over S at y. The following conditions are equivalent: (i) f is smooth at x. (ii) X is smooth over S at x and the morphism f ∗ ΩY /S → ΩX/S satisfies the equivalent conditions in 2.5.6 at x. Proof. (i)⇒(ii) follows from 2.4.1 (ii) and 2.5.4. (ii)⇒(i) The cokernel of f ∗ ΩY /S → ΩX/S is isomorphic to ΩX/Y . So ΩX/Y is free in a neighborhood of x. Shrinking X and Y , we may assume X and Y are smooth over S and ΩX/Y is a free OX -module with basis dg1 , . . . , dgn for some g1 , . . . , gn ∈ Γ(X, OX ). Let π : AnY → Y be the projection, and let g : X → AnY = Spec OY [t1 , . . . , tn ]

be the Y -morphism defined by the OY -algebra morphism OY [t1 , . . . , tn ] → f∗ OX ,

By 2.5.4, the sequence

ti 7→ gi

(i = 1, . . . , n).

0 → π ∗ ΩY /S → ΩAnY /S → ΩAnY /Y → 0

is exact. As ΩAnY /Y is free, the above sequence is split locally, and hence the sequence 0 → g ∗ π ∗ ΩY /S → g ∗ ΩAnY /S → g ∗ ΩAnY /Y → 0

is also exact. We have a commutative diagram

0 → g ∗ π ∗ ΩY /S → g ∗ ΩAnY /S → g ∗ ΩAnY /Y → 0 ∼ ↓ ↓ =↓ ∗ 0 → f ΩY /S → ΩX/S → ΩX/Y → 0.

After shrinking X, the sequence on the second line is exact by our assumption. By the choice of g, the last vertical arrow g ∗ ΩAnY /Y → ΩX/Y is an isomorphism. By the five lemma, we have g ∗ ΩAnY /S ∼ = ΩX/S . By 2.5.3, g is etale. So f is smooth. Theorem 2.5.8. Let S be a scheme, f : X → S a smooth morphism, i : Y → X a closed immersion defined by a quasi-coherent OX -ideal I locally of finite type, x a point of Y , n and m the relative dimensions of X and Y respectively at x, and n k : Am S = Spec OS [t1 , . . . , tm ] → AS = Spec OS [t1 , . . . , tn ]

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the closed immersion defined by the OS -algebra epimorphism tj if 1 ≤ j ≤ m, OS [t1 , . . . , tn ] → OS [t1 , . . . , tm ], tj 7→ 0 if m + 1 ≤ j ≤ n. The following conditions are equivalent: (i) f i : Y → S is smooth at x. (ii) There exists an open neighborhood U of x in X admitting an etale S-morphism g : U → AnS = Spec OS [t1 , . . . , tn ] such that the closed immersion iU : Y ∩U → U induced by i can be obtained from the closed immersion n k : Am S → AS by the base change g, that is, we have a Cartesian diagram i

U Y ∩U → U 0 g ↓ ↓g

k

→ AnS . Am S

(iii) There exist generators gm+1 , . . . , gn of the ideal Ix of OX,x such that dgm+1 , . . . , dgn form a part of a basis of (ΩX/S )x , or equivalently, the images of dgm+1 , . . . , dgn in (ΩX/S )x ⊗OX,x k(x) are linearly independent. (iv) The canonical morphism δ : i∗ (I /I 2 ) → i∗ ΩX/S satisfies the equivalent conditions in 2.5.6 at x. Proof. (i)⇒(ii) We have an exact sequence δ

i∗ (I /I 2 ) → i∗ ΩX/S → ΩY /S → 0.

Since X and Y are smooth over S at x, i∗ ΩX/S and ΩY /S are free in a neighborhood x. Choose g1 , . . . , gm ∈ OX,x so that the images of dg1 , . . . , dgm in (ΩY /S )x ⊗OY,x k(x) form a basis. Choose gm+1 , . . . , gn ∈ Ix so that the images of dg1 , . . . , dgn in (ΩX/S )x ⊗OX,x k(x) form a basis. Shrinking X, we may assume gm+1 , . . . , gn ∈ Γ(X, I ) and g1 , . . . , gn ∈ Γ(X, OX ). The OS -algebra morphisms OS [t1 , . . . , tn ] → f∗ OX ,

tj 7→ gj

OS [t1 , . . . , tm ] → (f i)∗ OY ,

tj 7→ gj |Y

(1 ≤ j ≤ n),

(1 ≤ j ≤ m),

define S-morphisms g : X → AnS and g 0 : Y → Am S such that the diagram i

Y → X g0 ↓ ↓g k

n Am S → AS

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commutes. Since ΩX/S is free in a neighborhood of x and the images of dg1 , . . . , dgn in (ΩX/S )x ⊗OX,x k(x) form a basis, the morphism g : X → AnS is etale at x by 2.5.3. Similarly the morphism g 0 : Y → Am S is also etale at x. Shrinking X, we may assume that these two morphisms are etale. 0 The morphisms i : Y → X and g 0 : Y → Am S induce a morphism i : Y → m 0 0 0 m X ×AnS AS such that π1 i = i and π2 i = g , where π1 : X ×AnS AS → X m and π2 : X ×AnS Am S → AS are the projections. Since π1 and i are closed immersions, i0 is a closed immersion. Since π2 and g 0 are etale, i0 is etale by 2.3.1 (iv). By 2.3.9, i0 : Y → X ×AnS Am S is an open and closed immersion. Shrinking X again, we may assume that it is an isomorphism. (ii)⇒(i) is obvious. (ii)⇒(iv) Let J be the ideal of OS [t1 , . . . , tn ] generated by tm+1 , . . . , tn . Since g is flat, we have I |U ∼ = g ∗ J . Since g is etale, we have g ∗ ΩAnS /S ∼ = (ΩX/S )|U ,

∼ g 0∗ ΩAm = (ΩY /S )|Y ∩U . S /S

One can check that the following sequence is exact: 0 → k ∗ (J /J 2 ) → k ∗ ΩAnS /S → ΩAm → 0. S /S Applying g 0∗ to this exact sequence, we get an exact sequence δ

0 → (i∗ (I /I 2 ))|Y ∩U → (i∗ ΩX/S )|Y ∩U → (ΩY /S )|Y ∩U → 0. The conditions of 2.5.6 thus hold for δ : i∗ (I /I 2 ) → i∗ ΩX/S at x. (iv)⇒(iii) Let gm+1 , . . . , gn ∈ Ix so that their images in Ix /Ix2 ⊗OX,x k(x) form a basis. By Nakayama’s lemma, gm+1 , . . . , gn generate Ix . Since Ix /Ix2 ⊗OX,x k(x) → (ΩX/S )x ⊗OX,x k(x) is injective, the images of dgm+1 , . . . , dgn in (ΩX/S )x ⊗OX,x k(x) form a basis. (iii)⇒(ii) Since ΩX/S is free in a neighborhood of x, we can find g1 , . . . , gm ∈ OX,x such that dg1 , . . . , dgn form a basis for (ΩX/S )x . Shrinking X, we may assume that g1 , . . . , gn ∈ Γ(X, OX ), gm+1 , . . . , gn ∈ Γ(X, I ), ΩX/S is free with basis dg1 , . . . , dgn , and gm+1 , . . . , gn generate I . The family g1 , . . . , gn defines an S-morphism g : X → AnS which is etale n by 2.5.3, and i : Y → X is induced from k : Am S → AS by the base change n g : X → AS . Corollary 2.5.9 (Jacobian Criterion). Let A be a ring, I a finitely generated ideal of A[t1 , . . . , tn ], and p a prime ideal of A[t1 , . . . , tn ] containing I. If there exist polynomials gm+1 , . . . , gn ∈ A[t1 , . . . , tn ] (0 ≤ m ≤ n)

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such that their images in the ideal Ip generate Ip and there exists a subset {im+1 , . . . , in } of {1, . . . , n} such that  ∂gm+1 ∂gm+1  ∂tim+1 · · · ∂tin  . ..   det  .  6∈ p,  .. ∂gn ∂gn · · · ∂ti ∂ti m+1

n

then Spec (A[t1 , . . . , tn ]/I) → Spec A is smooth at p/I. Furthermore, if m = 0, then Spec (A[t1 , . . . , tn ]/I) → Spec A is etale at p/I. Proof.

Use the equivalence of (i) and (iii) in 2.5.8.

Proposition 2.5.10. Consider a Cartesian diagram g0

X 0 = X ×Y Y 0 → X f0 ↓ ↓f g Y 0 → Y. Suppose that f : X → Y is locally of finite presentation and g : Y 0 → Y is faithfully flat. If f 0 is unramified (resp. etale, resp. smooth), then so is f . Proof. The statement for unramified morphisms is 2.2.5. Suppose f 0 is smooth. Let us prove f is smooth. The problem is local. We may assume that X = Spec B and Y = Spec A are affine, and B ∼ = A[t1 , . . . , tn ]/I for some finitely generated ideal I of A[t1 , . . . , tn ]. We have a closed immersion i : X → AnY with ideal sheaf I ∼ . Since g is flat, we have a closed immersion i0 : X 0 → AnY 0 with ideal sheaf g 00∗ I ∼ , where g 00 : AnY 0 → AnY is obtained from g : Y 0 → Y by base change. Since f 0 is smooth, by 2.5.8 (iv), we have an exact sequence 0 → i0∗ g 00∗ (I/I 2 )∼ → i0∗ ΩAnY 0 /Y 0 → ΩX 0 /Y 0 → 0 and ΩX 0 /Y 0 is locally free. Since g is faithfully flat, the sequence 0 → i∗ (I/I 2 )∼ → i∗ ΩAnY /Y → ΩX/Y → 0 is exact. Since f is locally of finite presentation, ΩX/Y is an OX -module locally of finite presentation. For any x ∈ X, let x0 ∈ X 0 be a preimage of x. Applying 1.6.6 to the quasi-compact faithfully flat morphism Spec OX 0 ,x0 → Spec OX,x and using the fact that (ΩX 0 /Y 0 )x0 is a free OX 0 ,x0 -module of finite rank, we see that (ΩX/Y )x is a free OX,x -module of finite rank. So ΩX/Y is locally free. By 2.5.8, f is smooth. The statement about etale morphisms follows from the statements about unramified morphisms and smooth morphisms since a morphism is etale if and only if it is unramified and smooth.

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([SGA 1] III, [EGA] IV 17.14.) Lemma 2.6.1. Let B be a ring, A and C two B-algebras, I an ideal of A with the property I 2 = 0, p : A → A/I the projection, and φ : C → A a B-homomorphism. Then there exists a one-to-one correspondence between the set {φ0 : C → A | φ0 is a B-homomorphism satisfying pφ0 = pφ} and the set HomA/I (ΩC/B ⊗C A/I, I), where A/I is regarded as a C-algebra through the homomorphism pφ : C → A/I. pφ

A/I ← C p ↑ . φ0 ↑ A ← B Proof. Since I 2 = 0, I can be regarded as an A/I-module and hence a Cmodule. Let φ0 : C → A be a B-homomorphism with the property pφ0 = pφ. Then the image of φ0 −φ is contained in I. We claim that φ0 −φ : C → I is a B-derivation. Indeed, since φ and φ0 are B-homomorphisms, the restriction of φ0 − φ to the image of B in C is 0. For any x1 , x2 ∈ C, we have (φ0 − φ)(x1 x2 ) = φ0 (x1 )φ0 (x2 ) − φ(x1 )φ(x2 )

= φ(x1 )(φ0 (x2 ) − φ(x2 )) + φ0 (x2 )(φ0 (x1 ) − φ(x1 )) = x1 · (φ0 (x2 ) − φ(x2 )) + x2 · (φ0 (x1 ) − φ(x1 )).

This proves our claim. Conversely, if D : C → I is a B-derivation, then one can verify φ + D : C → A is a B-homomorphism with the property p(φ + C) = pφ. Our assertion then follows from the fact that the set of B-derivations DerB (C, I) is in one-to-one correspondence with the set HomC (ΩC/B , I) ∼ = HomA/I (ΩC/B ⊗C A/I, I). Theorem 2.6.2. Let f : X → S be a morphism locally of finite presentation. The following conditions are equivalent: (i) f is unramified (resp. etale, resp. smooth). (ii) For any S-scheme Spec A and any ideal I of A with the property I 2 = 0, the canonical map HomS (Spec A, X) → HomS (Spec A/I, X) is injective (resp. bijective, resp. surjective).

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Proof. (i)⇒(ii) We have one-to-one correspondences HomS (Spec A, X) ∼ = HomSpec A (Spec A, Spec A ×S X), ∼ HomS (Spec A/I, X) = HomSpec A (Spec A/I, Spec A ×S X). It suffices to show that HomSpec A (Spec A, Spec A ×S X) → HomSpec A (Spec A/I, Spec A ×S X) is injective (resp. bijective, resp. surjective). Making the base change Spec A → S, we are reduced to prove that if f : X → Spec A is unramified (resp. etale, resp. smooth), then the canonical map HomSpec A (Spec A, X) → HomSpec A (Spec A/I, X) is injective (resp. bijective, resp. surjective). Suppose f : X → Spec A is unramified. An element g in HomSpec A (Spec A, X) is a section of f and hence is an open immersion. (Confer the proof of 2.3.10.) We have g = (f |im g )−1 . So g is completely determined by im g. Let g 0 be the composite g

Spec A/I → Spec A → X. Since I 2 = 0, Spec A/I and Spec A have the same underlying topological space. So g and g 0 have the same image in X. It follows that g is completely determined by g 0 , and the map HomSpec A (Spec A, X) → HomSpec A (Spec A/I, X) is injective. The statement for etale morphisms follows from 2.3.12. Suppose f : X → Spec A is smooth. Let {Uα } be a covering of X by affine open subsets such that we have etale A-morphisms gα : Uα → AnAα . Let i : Spec A/I → Spec A be the closed immersion. Given an A-morphism g 0 : Spec A/I → X, g0

Spec A/I → X i↓ ↓ Spec A = Spec A, regard each g 0−1 (Uα ) as an open subset of Spec A, and cover it by affine open 0 subschemes Vαβ of Spec A. Let gαβ : Vαβ ⊗A A/I → Uα be the morphisms 0 induced by g , and let i : Vαβ ⊗A A/I → Vαβ be the closed immersions. 0 We can construct A-morphisms hαβ : Vαβ → AnA such that hαβ i = gα gαβ .

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Since gα is etale, by the etale case that we have just treated, there exist morphisms gαβ : Vαβ → Uα such that the following diagram commutes: 0 gαβ

Vαβ ⊗A A/I → Uα gαβ

i

↓

Vαβ

% ↓ gα

hαβ

→ AnAα .

For convenience, we change our notation on indices: We have shown that there exist a covering {Uλ } (resp. {Vλ }) of X (resp. Spec A) by affine open subsets and A-morphisms gλ : Vλ → Uλ such that g 0 (Vλ ⊗A A/I) ⊂ Uλ , and gλ i = gλ0 , where i : Vλ ⊗A A/I → Vλ are the closed immersions, and gλ0 : Vλ ⊗A A/I → Uλ the morphisms induced by g 0 . For each pair (λ, µ) of indices, consider the commutative diagram 0 gλµ

(Vλ ∩ Vµ ) ⊗A A/I → Uλ ∩ Uµ gλµ

↓ Vλ ∩ Vµ

% ↓f → Spec A,

i

where 0 gλµ = gλ0 |(Vλ ∩Vµ )⊗A A/I = gµ0 |(Vλ ∩Vµ )⊗A A/I ,

gλµ = gλ |Vλ ∩Vµ .

If we replace gλµ in this diagram by gµλ = gµ |Vλ ∩Vµ , the diagram still commutes. By 2.6.1, gµλ corresponds to an element sλµ ∈ Γ((Vλ ∩ Vµ ) ⊗A A/I, H omOSpec A/I (g 0∗ ΩX/Spec A , I ∼ )). One can check that on (Vλ ∩ Vµ ∩ Vν ) ⊗A A/I, we have sλµ + sµν + sνλ = 0. Since Spec A/I is affine, we have ˇ 1 (Spec A/I, H omO H (g 0∗ ΩX/Spec A , I ∼ )) = 0. Spec A/I Replacing {Vλ } by a refinement, we may assume there exist

sλ ∈ Γ(Vλ ⊗A A/I, H omOSpec A/I (g 0∗ ΩX/Spec A , I ∼ ))

such that sλµ = sλ − sµ . Consider the commutative diagram g0

λ Vλ ⊗A A/I →

↓ Vλ i

gλ

Uλ

% ↓f → Spec A.

By 2.6.1, each sλ corresponds to an A-morphism g˜λ : Vλ → Uλ such that g˜λ i = gλ0 . One can check that g˜λ and g˜µ coincide on Vλ ∩ Vµ . So we can

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glue g˜λ together to get an A-morphism g : Spec A → X such that gi = g 0 . This proves that the map HomSpec A (Spec A, X) → HomSpec A (Spec A/I, X) is surjective. (ii)⇒(i) The problem is local. We may assume that X = Spec C and S = Spec B are affine. Suppose that for any B-algebra A and any ideal I of A satisfying I 2 = 0, the canonical map HomSpec B (Spec A, Spec C) → HomSpec B (Spec A/I, Spec C) is injective. Then by 2.6.1, we have HomA/I (ΩC/B ⊗C A/I, I) = 0 whenever there exists a B-homomorphism from C to A. Take A = (C ⊗B C)/J 2 and I = J/J 2 , where J is the kernel of the epimorphism C ⊗B C → C,

x ⊗ y 7→ xy.

We have a B-algebra homomorphism C → A,

x 7→ x ⊗ 1 + J 2 .

So HomA/I (ΩC/B ⊗C A/I, I) = 0, that is, HomC (ΩC/B , ΩC/B ) = 0. Hence ΩC/B = 0 and f is unramified. Suppose for any B-algebra A and any ideal I of A satisfying I 2 = 0, the canonical map HomSpec B (Spec A, Spec C) → HomSpec B (Spec A/I, Spec C) is surjective. We may assume C = B[t1 , . . . , tn ]/J for some finitely generated ideal J of B[t1 , . . . , tn ]. Taking A = B[t1 , . . . , tn ]/J 2 and I = J/J 2 , we see that the map HomB (B[t1 , . . . , tn ]/J, B[t1 , . . . , tn ]/J 2 ) → HomB (B[t1 , . . . , tn ]/J, B[t1 , . . . , tn ]/J) is surjective. So there exists a B-homomorphism φ : B[t1 , . . . , tn ]/J → B[t1 , . . . , tn ]/J 2 which is a section of the projection p : B[t1 , . . . , tn ]/J 2 → B[t1 , . . . , tn ]/J. Consider id − φp. As p(id − φp) = p − (pφ)p = p − p = 0,

the image of id − φp lies in J/J 2 . We claim that

id − φp : B[t1 , . . . , tn ]/J 2 → J/J 2

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is a B-derivation. Indeed, we have (id − φp)(b) = 0 for any b ∈ B since φ is a B-homomorphism. For any x1 , x2 ∈ B[t1 , . . . , tn ]/J 2 , we have (id − φp)(x1 x2 )

= x1 x2 − φp(x1 )φp(x2 )

= x1 (x2 − φp(x2 )) + (x1 − φp(x1 ))φp(x2 )

= x1 (x2 − φp(x2 )) + x2 (x1 − φp(x1 )) − (x1 − φp(x1 ))(x2 − φp(x2 )) = x1 (x2 − φp(x2 )) + x2 (x1 − φp(x1 )).

This proves our claim. Let q : B[t1 , . . . , tn ] → B[t1 , . . . , tn ]/J 2 be the projection. Then (id − φp)q : B[t1 , . . . , tn ] → J/J 2 is a B-derivation. It corresponds to a (B[t1 , . . . , tn ]/J)-module homomorphism ψ : ΩB[t1 ,...,tn ]/B ⊗B[t1 ,...,tn ] (B[t1 , . . . , tn ]/J) → J/J 2 . We claim that ψ is a left inverse of the canonical homomorphism δ : J/J 2 → ΩB[t1 ,...,tn ]/B ⊗B[t1 ,...,tn ] (B[t1 , . . . , tn ]/J). Indeed, for any x ∈ J, we have ψδ(x + J 2 ) = ψ(dx ⊗ 1) = (id − φp)q(x) = x since pq|J = 0. This proves our claim. So the sequence 0 → J/J 2 → ΩB[t1 ,...,tn ]/B ⊗B[t1 ,...,tn ] (B[t1 , . . . , tn ]/J) → Ω(B[t1 ,...,tn ]/J)/B → 0

is split exact. By 2.5.8 (iv), Spec B[t1 , . . . , tn ]/J → Spec B is smooth. The statement for the etale morphism follows from that for the unramified and smooth morphism. 2.7

Direct Limits and Inverse Limits

Let I and C be two categories, and let F : I → C be a covariant (resp. contravariant) functor. We often write F as (F (i))i∈ob I , or simply (F (i)). For every object X in C , let CX : I → C be the constant functor that

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maps each object in I to X, and each morphism in I to idX . Consider the covariant (resp. contravariant) functor C → (Sets), X 7→ Hom(F, CX ) (resp. C → (Sets), X 7→ Hom(CX , F )) from C to the category of sets. If this functor is representable, we call the object in C representing this functor the direct limit (resp. inverse limit) of F , and denote it by limi∈ob I F (i) (resp. limi∈ob I F (i)). We can also −→ ←− define limi∈ob I F (i) (resp. limi∈ob I F (i)) as an object in C together with −→ ←− morphisms φi : F (i) → lim F (i) (resp. φi : lim F (i) → F (i)) −→ ←− i∈ob I

i∈ob I

with the property φj ◦ F (fij ) = φi

(resp. F (fij ) ◦ φj = φi )

whenever there exists a morphism fij : i → j in I. Moreover, these data satisfy the following universal property: For any object X in C together with morphisms ψi : F (i) → X (resp. ψi : X → F (i)) with the property ψj ◦ F (fij ) = ψi (resp. F (fij ) ◦ ψj = ψi ) whenever there exists a morphism fij : i → j in I, there exists a unique morphism ψ : limi∈ob I F (i) → X −→ (resp. ψ : X → limi∈ob I F (i)) in C such that ψφi = ψi (resp. φi ψ = ψi ). ←− In the following, C is the category of sets, or the category of groups. Suppose objects in I form a set. Then for any contravariant functor Q F : I → C , limi F (i) exists. In fact, limi F (i) is the subspace of i∈ob I F (i) ←− ← Q− consisting of those elements (xi ) ∈ i∈ob I F (i) such that F (fij )(xj ) = xi whenever there exists a morphism fij : i → j. The canonical morphisms Q φi : limi F (i) → F (i) are induced by the projections i F (i) → F (i). ←− Consider the following conditions of the category I: (I1) Given two morphisms i → j and i → j 0 in I, there exist two morphisms j → k and j 0 → k such that the following diagram commutes: i → j ↓ ↓ 0 j → k. (I2) Given two morphisms i ⇒ j in I, there exists a morphism j → k so that its composite with these two morphisms are the same. (I3) Given two objects i and j in I, there exists an object k in I admitting morphisms i → k and j → k. Note that (I2) + (I3) ⇒ (I1).

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Suppose objects in I form a set and I satisfies (I2) and (I3). For any covariant functor F : I → C , limi F (i) exists. It can be defined as the set −→ (or group) a F (i) /R, i∈I

` where R is the equivalence relation on i∈I F (i) defined as follows: For any x ∈ F (i) and y ∈ F (j), we say x ∼ y if there exist morphisms i → k and j → k in I such that x and y have the same image in F(k). In the case ` where C is the category of groups, the group structure on i∈I F (i) /R

is defined as follows: For any x ∈ F (i) and y ∈ F (j), let i → k and j → k be morphisms in I. Then xy is defined to be the product of the images of x and y in F (k), and passed to the quotient. The canonical morphisms ` F (i) → limi F (i) are induced by the inclusions F (i) → i∈I F (i). Note −→ that limI is an exact functor, that is, if F → G → H is a sequence of −→ functors such that are exact for all i, then

lim F (i) → lim G(i) → lim H(i) −→ −→ −→ i

is exact.

F (i) → G(i) → H(i)

i

i

Suppose objects in I form a set and I satisfies (I1) and (I2). Define an equivalence relation on I as follows: We say that i ∼ j if there exists a sequence of objects i = i 0 , i 1 , . . . , in = j in I such that for each 0 ≤ m ≤ n − 1, there exists a morphism im → im+1 or a morphism im+1 → im . Let Iλ (λ ∈ Λ) be the equivalence classes of I with respect to this relation. Then each Iλ satisfies (I2) and (I3). For any covariant functor F : I → C , limi∈ob I F (i) exists, and −→ (` limi∈ob I F (i) if C = (Sets), λ∈Λ −→ λ lim F (i) ∼ = L −→ F (i) if C = (Groups). λ∈Λ lim i∈ob I −→i∈ob Iλ limI is an exact functor. −→ Assume that objects in I form a set. For any covariant functor F : I → C , limi∈ob I F (i) exists, and −→  `  F (i) /R if C = (Sets), ∼ L i∈ob I lim F (i) = −→ 0  if C = (Groups), i∈ob I i∈ob I F (i) /R

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where R (resp. R0 ) is the equivalence relation generated by x ∼ y (resp. the subgroup generated by x − y), where x ∈ F (i), Y ∈ F (j), and there exists a morphism i → j such that the image of x in F (j) is y. Let J be a full subcategory of I. We say that J is cofinal in I if for any object i in I, there exists a morphism i → j in I such that j is an object in J. If I satisfies (I1) (resp. (I2), resp. (I3)) and J is cofinal in I, then J satisfies the same condition. Suppose I satisfies (I1) and J is cofinal in I, then for any covariant (resp. contravariant) functor F : I → C and any object X in C , the canonical map Hom(F, CX ) → Hom(F |J , CX |J ) (resp. Hom(CX , F ) → Hom(CX |J , F |J )) is bijective. So we have lim F (i) ∼ F (j) = lim −→ −→

i∈ob I

(resp.

j∈ob J

lim F (i) ∼ F (j)). = lim ←− ←−

i∈ob I

j∈ob J

If I satisfies (I1) and has a cofinal full subcategory J whose objects form a set, we can apply our previous descriptions of limj∈ob J F (j) and ←− limj∈ob J F (j) to get descriptions of limi∈ob I F (i) and limi∈ob I F (i). An −→ ←− −→ object j in I is called a final object if for any object i in I, there exists one and only one morphism i → j. In this case, the category J consisting of only one object j and only one morphism idj is a cofinal full subcategory of I, and we have F (j) ∼ F (i) = lim −→

(resp. lim F (i) ∼ = F (j).) ←−

i∈ob I

2.8

i∈ob I

Henselization

([EGA IV] 18.5–8.) Lemma 2.8.1. (i) Let A be a ring. The map a 7→ D(a) defines a one-to-one correspondence between the set of idempotent elements in A and the set of open and closed subsets of Spec A. (ii) Let A be a ring. The map a 7→ (Ann(a), Ann(1 − a))

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defines a one-to-one correspondence between the set of idempotent elements in A and the set of pairs (a1 , a2 ) of ideals of A with the properties a1 ∩ a2 = 0,

a1 + a2 = A.

(iii) Let (R, m) be a local ring, let f (t) ∈ R[t] be a monic polynomial, and let A = R[t]/(f (t)). Then the map (g(t), h(t)) 7→ (Ag(t), Ah(t)) defines a one-to-one correspondence between the set of pairs of monic polynomials (g(t), h(t)) in R[t] with the properties f (t) = g(t)h(t),

g(t)R[t] + h(t)R[t] = R[t],

and the set of pairs (a1 , a2 ) of ideals of A with the properties a1 ∩ a2 = 0,

a1 + a2 = A.

Proof. (i) If a ∈ A is idempotent, then we have a(1 − a) = 0,

a + (1 − a) = 1.

It follows that D(a) ∩ D(1 − a) = Ø,

D(a) ∪ D(1 − a) = Spec A.

So D(a) is an open and closed subset of Spec A. Suppose that U is an open and closed subset of A. Let a ∈ A be the element corresponding to the section in Γ(Spec A, OSpec A ) such that a|U = 1 and a|Spec A−U = 0. Then a is idempotent and U = D(a). So the map a 7→ D(a) is surjective. To prove it is injective, let a1 and a2 be idempotent elements in A such that D(a1 ) = D(a2 ), and let us prove that a1 = a2 . It suffices to show that for any prime ideal p of A, a1 and a2 have the same image in Ap . So we may assume that A is a local ring. Then Spec A is connected. It follows that either D(a1 ) = D(a2 ) = Ø or D(a1 ) = D(a2 ) = Spec A. In the first case, a1 and a2 are nilpotent. Since they are idempotent, we have a1 = a2 = 0. In the second case, we have D(1 − a1 ) = D(1 − a2 ) = Ø, and the same argument shows that 1 − a1 = 1 − a2 = 0. So we always have a1 = a2 .

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(ii) For any a ∈ A, if x ∈ Ann(a) ∩ Ann(1 − a), then x = x · (a + (1 − a)) = 0. If a is idempotent, then a(1−a) = 0. So 1−a ∈ Ann(a) and a ∈ Ann(1−a), and hence 1 = (1 − a) + a ∈ Ann(a) + Ann(1 − a). It follows that Ann(a) ∩ Ann(1 − a) = 0,

Ann(a) + Ann(1 − a) = A.

Suppose that a1 and a2 are idempotent elements in A such that Ann(a1 ) = Ann(a2 ). We have 1 − a1 ∈ Ann(a1 ). It follows that (1 − a1 )a2 = 0, that is, a2 = a1 a2 . Similarly, we have a1 = a1 a2 . So a1 = a2 . Thus the map a 7→ (Ann(a), Ann(1 − a)) is injective. Suppose that (a1 , a2 ) is a pair of ideals of A with the properties a1 ∩a2 = 0 and a1 + a2 = A. By the Chinese remainder theorem, we have A∼ = A/a1 × A/a2 . Note that Ann(1, 0) = {0} × A/a2 ,

Ann(0, 1) = A/a1 × {0}.

Let a be the element in A corresponding to the element (1, 0) in A/a1 ×A/a2 . Then we have (Ann(a), Ann(1 − a)) = (a1 , a2 ). So the map a 7→ (Ann(a), Ann(1 − a)) is surjective. (iii) Suppose that (g(t), h(t)) is a pair of monic polynomials in R[t] with the properties f (t) = g(t)h(t) and g(t)R[t] + h(t)R[t] = R[t]. Then Ag(t) + Ah(t) = A. Choose polynomials a(t), b(t) ∈ R[t] such that a(t)g(t) + b(t)h(t) = 1. If r(t) ∈ g(t)R[t] ∩ h(t)R[t], then we have r(t) = g(t)p(t) = h(t)q(t) for some polynomials p(t) and q(t). We then have r(t) = a(t)g(t)r(t) + b(t)h(t)r(t) = a(t)g(t)h(t)q(t) + b(t)h(t)g(t)p(t) = f (t)(a(t)q(t) + b(t)p(t)).

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So r(t) ∈ f (t)R[t]. It follows that Ag(t) ∩ Ah(t) = 0. Suppose (g1 (t), h1 (t)) is another pair of monic polynomials in R[t] with the properties f (t) = g1 (t)h1 (t) and g1 (t)R[t] + h1 (t)R[t] = R[t]. If Ag(t) = Ag1 (t), then g(t)R[t] = g1 (t)R[t]. Since g(t) and g1 (t) are monic, this implies that g(t) = g1 (t). Similarly h(t) = h1 (t). Given a pair (a1 , a2 ) of ideals of A with the properties a1 ∩ a2 = 0 and a1 + a2 = A, let A1 = A/a1 and A2 = A/a2 . By the Chinese remainder theorem, we have R[t]/(f (t)) ∼ = A1 × A2 .

Let u be the image of t in A1 . Then A1 /mA1 is generated by the image u ¯ of u in A1 /mA1 as a k-algebra, where k = R/m. Let g¯(t) ∈ k[t] be the minimal polynomial of u¯, and let d = deg g¯(t). Then {1, u ¯, . . . , u¯d−1 } is a basis of A1 /mA1 over k. By Nakayama’s lemma, A1 is generated by 1, u, . . . , ud−1 as an R-algebra. So there exists a monic polynomial g(t) ∈ R[t] of degree d such that g(u) = 0. g(t) is necessarily a lifting of g¯(t). Since A1 is a direct factor of the free R-module R[t]/(f (t)) and R is a local ring, A1 is a free R-module. As {1, u ¯, . . . , u¯d−1 } is a basis of A1 /mA1 over k, {1, u, . . . , ud−1 } is a basis of A1 over R. So the R-algebra homomorphism R[t]/(g(t)) → A1 ,

t 7→ u

is an isomorphism. This is equivalent to saying that a1 = Ag(t). Similarly, there exists a monic polynomial h(t) ∈ R[t] such that a2 = Ah(t). We have R[t]/(f (t)) ∼ = R[t]/(g(t)) × R[t]/(h(t)). The image of g(t)h(t) in R[t]/(g(t)) × R[t]/(h(t)) is 0. So its image in R[t]/(f (t)) is 0, that is, f (t)|g(t)h(t). On the other hand, we have deg f (t) = rank R[t]/(f (t)) = rank R[t]/(g(t)) × R[t]/(h(t)) = deg (g(t)h(t)).

As f (t) and g(t)h(t) are monic polynomials, we have f (t) = g(t)h(t). So we have R[t]/(g(t)h(t)) ∼ = R[t]/(g(t)) × R[t]/(h(t)).

It is clear that the kernels of the projections

R[t]/(g(t)) × R[t]/(h(t)) → R[t]/(g(t)),

R[t]/(g(t)) × R[t]/(h(t)) → R[t]/(h(t))

generate the ideal R[t]/(g(t))×R[t]/(h(t)). So the kernels of the projections R[t]/(g(t)h(t)) → R[t]/(g(t)), R[t]/(g(t)h(t)) → R[t]/(h(t))

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generate the ideal R[t]/(g(t)h(t)), that is, (g(t))/(g(t)h(t)) and (h(t))/(g(t)h(t)) generate the ideal R[t]/(g(t)h(t)). This implies that g(t) and h(t) generate the ideal R[t]. Lemma 2.8.2. Let (R, m) be a local ring and let A be an R-algebra which is free of finite rank as an R-module. For any R-algebra A0 , let Idem(A0 ⊗R A) be the set of idempotent elements in A0 ⊗R A. Then there exists an etale Ralgebra B of finite presentation such that the functor A0 7→ Idem(A0 ⊗R A) is represented by B, that is, we have one-to-one correspondences Idem(A0 ⊗R A) ∼ = HomR (B, A0 ) functorial in A0 . Proof.

Let {e1 , . . . , en } be a basis of A over R. We have X ei ej = aijk ek k

for some aijk ∈ R. Let

Pk (t1 , . . . , tn ) =

X i,j

aijk ti tj − tk

(k = 1, . . . , n).

P An element i ai ei (ai ∈ R) in A is idempotent if and only if Pk (a1 , . . . , an ) = 0 for all k. For any R-algebra A0 , {1 ⊗ e1 , . . . , 1 ⊗ en } P is a basis of A0 ⊗R A over A0 . An element i a0i ⊗ ei (a0i ∈ A0 ) in A0 ⊗R A is idempotent if and only if Pk (a01 , . . . , a0n ) = 0 for all k. So we have a one-to-one correspondence Idem(A0 ⊗R A) ∼ = HomR (R[t1 , . . . , tn ]/(P1 , . . . , Pn ), A0 ). Let us prove R[t1 , . . . , tn ]/(P1 , . . . , Pn ) is etale over R. By 2.6.2, it suffices to show for any R-algebra A0 and any ideal I of A0 satisfying I 2 = 0, that the canonical map HomR (R[t1 , . . . , tn ]/(P1 , . . . , Pn ), A0 ) → HomR (R[t1 , . . . , tn ]/(P1 , . . . , Pn ), A0 /I) is bijective, that is, the map

Idem(A0 ⊗R A) → Idem (A0 /I) ⊗R A

is bijective. This is equivalent to saying that we have a one-to-one correspondence between the set of open and closed subsets ofSpec (A0 ⊗R A) and the set of open and closed subsets of Spec (A0 /I) ⊗R A . This follows from 0 the fact that Spec (A ⊗R A) has the same underlying topological space as 0 Spec (A /I) ⊗R A

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Proposition 2.8.3. Let (R, m) be a local ring, k = R/m its residue field, S = Spec R, and s the closed point of S. The following conditions are equivalent. (i) Every finite R-algebra A is a direct product of local rings. (ii) The condition (i) holds for A = R[t]/(f (t)) for any monic polynomial f (t) ∈ R[t]. (iii) For any finite R-algebra A, the canonical homomorphism A → A/mA induces a one-to-one correspondence between the set of idempotent elements in A and the set of idempotent elements in A/mA. (iv) The condition (iii) holds for A = R[t]/(f (t)) for any monic polynomial f (t) ∈ R[t]. (v) For any monic polynomial f (t) ∈ R[t] and any factorization f¯(t) = ¯ ¯ g¯(t)h(t), where f¯(t) is the image of f (t) in k[t], and g¯(t) and h(t) are relatively prime monic polynomials in k[t], there exist uniquely determined polynomials g(t) and h(t) in R[t] such that f (t) = g(t)h(t), g¯(t) and ¯h(t) are images of g(t) and h(t) in k[t], respectively, and the ideal generated by g(t) and h(t) is R[t]. ¯ such (vi) The condition (v) holds for any factorization f¯(t) = (t− a ¯)h(t) ¯ that t − a ¯ and h(t) are relatively prime monic polynomials in k[t]. (vii) For any etale morphism g : X → S, any section of gs : X ⊗R k → Spec k is induced by a section g. When (R, m) satisfies the above equivalent conditions, we say R is henselian. If R is henselian and k = R/m is separably closed, we say R is strictly local or strictly henselian. Proof. (i)⇒(iii) Let A be a finite R-algebra. Any maximal ideal of A lies over the maximal ideal m of R. Since A/mA is finite over R/m, it is artinian and has finitely many maximal ideals. It follows that A has finitely many maximal ideals. For any maximal ideal n of A, the canonical morphism Spec An → Spec A is an embedding of topological spaces and Spec An is connected. When n goes over the finite set of maximal ideals of A, the images of Spec An → Spec A cover Spec A. So Spec A has finitely many connected components. Similarly, Spec A/mA has finitely many connected components. Note that if a topological space has finitely many connected components, then open and closed subsets are exactly unions of connected components. By 2.8.1 (i), to prove (iii), it suffices to show that taking inverse image under the morphism Spec A/mA → Spec A defines a one-toone correspondence between the set of open and closed subsets of Spec A

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and the set of open and closed subsets of Spec A/mA. By assumption, we have Y A∼ An . = n

So we have

Spec A ∼ =

a

Spec An .

n

Each Spec An can be regarded as a connected open and closed subset of Spec A. So any open closed subset of Spec A is a union of some Spec An . On the other hand, we have Y a Spec A/mA ∼ Spec An /mAn , = Spec ( An /mAn ) ∼ = n

n

and each Spec An /mAn consists of only one point. Any open and closed subset of Spec A/mA is a union of some Spec An /mAn . So taking inverse image under the morphism Spec A/mA → Spec A defines a one-to-one correspondence between the set of open and closed subsets of Spec A and the set of open and closed subsets of Spec A/mA. (iii)⇒(i) If a topological space has finitely many connected components, then connected components are exactly minimal open and closed subsets. So taking inverse images under the morphism Spec A/mA → Spec A defines a one-to-one correspondence between the set of connected components of Spec A and the set of connected components of Spec A/mA. Note that Spec A/mA is discrete and its connected components are of the form {n/mA} for maximal ideals n of A. So different maximal ideals of A lie in different connected components of Spec A. Let n1 and n2 be two different maximal ideals of A. We claim that Spec An1 and Spec An2 are disjoint considered as subsets of Spec A. Indeed, if they have a nonempty intersection, then Spec An1 ∪ Spec An2 is connected. This contradicts that the fact that n1 and n2 lie in different connected components. So as a set, Spec A is a disjoint union of Spec An . One can show that each Spec An is a maximal connected subset of Spec A. So each Spec An is a connected component of ` Q Spec A. It follows that Spec A = n Spec An as schemes. So A = n An . (i)⇒(ii) Trivial. (ii)⇒(iv) Use the same argument as (i)⇒(iii). (iv)⇒(iii) Let e1 , e2 ∈ A be idempotent elements such that e1 ≡ e2 mod mA. We have (e1 − e2 )3 = e31 − 3e21 e2 + 3e1 e22 − e32 = e1 − 3e1 e2 + 3e1 e2 − e2 = e1 − e2 .

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So we have (e1 − e2 )((e1 − e2 )2 − 1) = 0. Since e1 − e2 ∈ mA, (e1 − e2 )2 − 1 is a unit in A. It follows that e1 − e2 = 0. So the map A → A/mA is always injective when restricted to the set of idempotent elements in A. Let e¯ ∈ A/mA be an idempotent element. We need to show it can be lifted to an idempotent element in A. Let a ∈ A be an arbitrary lift of e¯, and let A0 = R[a]. We claim that e¯ is the image of an idempotent element e¯0 in A0 /mA0 . Let a ¯ be the image of a in A0 /mA0 and let f : Spec A/mA → Spec A0 /mA0 be the canonical morphism. Then f −1 (D(¯ a)) = D(¯ e). Since Spec A0 /mA0 is discrete, D(¯ a) is open and closed. So there exists an idempotent element e¯0 in A0 /mA0 such that D(¯ e0 ) = D(¯ a). We have f −1 (D(¯ e0 )) = D(¯ e). This implies that the image of e¯0 in A/mA is e¯. To prove (ii), it suffices to show that e¯0 can be lifted to an idempotent element in A0 . Replacing A by A0 , we are reduced to the case where A = R[a]. Since a is integral over R, there exists a monic polynomial f (t) ∈ R[t] such that f (a) = 0. We then have an epimorphism of R-algebras R[t]/(f (t)) → A,

t 7→ a.

We claim that e¯ is the image of an idempotent element in k[t]/(f¯(t)), where f¯(t) is the image of f (t) in k[t]. Indeed, the homomorphism k[t]/(f¯(t)) → A/mA is surjective. The corresponding morphism Spec A/mA → Spec k[t]/(f¯(t))

is a closed immersion. Since Spec k[t]/(f¯(t)) is discrete, the open and closed subset D(¯ e) of Spec A/mA is also open and closed in Spec k[t]/(f¯(t)). So there exists an idempotent element e¯00 in k[t]/(f¯(t)) such that D(¯ e) = D(¯ e00 ). The image e¯00 in A/mA is e¯. If the condition (iv) holds, there exists an idempotent element e00 in R[t]/(f (t)) lifting e¯00 . Then the image e of e00 in A is an idempotent element lifting e¯. (iv)⇒(v) Use 2.8.1 (ii), (iii).

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(v)⇒(vi) Trivial. (vi)⇒(vii) Given a section of gs , let x be its image. By 2.3.5, there exists a monic polynomial f (t) ∈ R[t] and a maximal ideal n of B = R[t]/(f (t)) not containing the image of f 0 (t) in B such that OX,x is R-isomorphic to Bn . Let f¯(t) be the image of f (t) in k[t]. The given section of gs defines a k-morphism Spec k → Spec OX,x /mOX,x . Composed with the canonical morphisms Spec OX,x /mOX,x ∼ = Spec Bn /mBn → Spec B/mB ∼ = Spec k[t]/(f¯(t)), we get a k-morphism Spec k → Spec k[t]/(f¯(t)). It corresponds to a k-algebra homomorphism k[t]/(f¯(t)) → k.

Let a ¯ be the image of t under this homomorphism. Then f¯(¯ a) = 0 and f¯0 (¯ a) 6= 0. So a ¯ is a simple root of f¯(t), and we have ¯ f¯(t) = (t − a ¯)h(t)

¯ for some monic polynomial h(t) ∈ k[t] relatively prime to t − a ¯. If the condition (vi) holds, this factorization can be lifted to a factorization f (t) = (t − a)h(t) for some a ∈ R lifting a ¯ and some monic polynomial h(t) ∈ R[t] lifting ¯ h(t) such that the ideal generated by t − a and h(t) is R[t]. The R-algebra homomorphism R[t]/(f (t)) → R,

t 7→ a

induces an R-algebra homomorphism Bn = (R[t]/(f (t)))n → R and hence an S-morphism S = Spec R → Spec Bn ∼ = Spec OX,x . Composed with the canonical morphism Spec OX,x → X, we get a section of g : X → S inducing the given section of gs . (vii)⇒(iv) Let B be the etale R-algebra in 2.8.2 for the free R-algebra of finite rank A = R[t]/(f (t)). In the proof of (iv)⇒(iii), we have shown that the map Idem(A) → Idem(A/mA)

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is always injective. We need to show it is surjective. It suffices to show that the canonical map HomR (B, R) → HomR (B, R/m)

is surjective. HomR (B, R) (resp. HomR (B, R/m)) can be identified with the set of sections of the morphism g : Spec B → Spec R (resp. gs : Spec (B ⊗R k) → Spec k). Since g is etale, our assertion follows if condition (vii) holds. Proposition 2.8.4. Suppose (R, m) is a complete local noetherian ring. Then R is henselian. Proof. First Method: Let us verify that the condition 2.8.3 (i) holds. Let A be a finite R-algebra. Then A is complete with respect to the m-adic topology. Any maximal ideal of A lies over the maximal ideal m of R. Since A/mA is finite over R/m, it is artinian. It follows that A has finitely many maximal ideals. Let m1 , . . . , mn be all the maximal ideals of A. Then the nilpotent radical of A/mA is (m1 ∩ · · · ∩ mn )/mA. So there exists a positive integer k such that (m1 ∩ · · · ∩ mn )k ⊂ mA ⊂ m1 ∩ · · · ∩ mn .

Combined with the Chinese remainder theorem, we get bm1 ×· · ·×A bmn . A∼ A/mi A ∼ A/mi1 · · · min ∼ (A/mi1 ×· · ·×A/min ) ∼ = lim = lim = lim =A ←− ←− ←− i

i

i

So A is a direct product of local rings. Second Method: Let us verify that the condition 2.8.3 (vi) holds. Let f (t) ∈ R[t] be a monic polynomial and let ¯ f¯(t) = (t − a ¯)h(t)

¯ be a factorization of the image f¯(t) of f (t) in k[t], where t − a ¯ and h(t) are relatively prime monic polynomials in k[t]. We lift a ¯ to a root of f (t) in R using a method similar to Newton’s iteration method of finding a root of a smooth function in calculus. Note that a ¯ is a simple root of f¯(t). So we have f¯(¯ a) = 0, f¯0 (¯ a) 6= 0. Let a0 ∈ R be an arbitrary lift of a ¯. Then f (a0 ) ∈ m and f 0 (a0 ) is a unit in R. Suppose that we have found a lift ai ∈ R of a ¯ such that f (ai ) ∈ mi+1 . Then f 0 (ai ) is a unit in R. Define f (ai ) ai+1 = ai − 0 . f (ai )

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We have ai+1 − ai = −

f (ai ) ∈ mi+1 . f 0 (ai )

So ai+1 is also a lift of a ¯. Note that for any a, δ ∈ R, we can find b ∈ R such that f (a + δ) = f (a) + f 0 (a)δ + bδ 2 . In particular, we can find bi ∈ R such that f (a ) 2 f (a ) i i + bi − 0 f (ai+1 ) = f (ai ) + f 0 (ai ) · − 0 f (ai ) f (ai ) f (a ) 2 i = bi − 0 ∈ m2i+2 ⊂ mi+2 . f (ai )

Since R is complete, the limit

a = lim ai i→∞

exists in R. It is a lift of a ¯, and f (a) = 0. So we have f (t) = (t − a)h(t)

¯ for some monic polynomial h(t) lifting h(t). Since the ideal generated by ¯ t−¯ a and h(t) is k[t], by Nakayama’s lemma applied to the finitely generated R-module R[t]/(f (t)), we see the ideal generated by t − a and h(t) is R[t]. Finally let us prove that the lift of a ¯ to a root of f (t) in R is unique. 0 Let a, a ∈ R be two roots of f (t) lifting a ¯. We can find b ∈ R such that f (a0 ) = f (a) + f 0 (a)(a0 − a) + b(a0 − a)2 ,

that is, a0 − a = −

b f 0 (a)

(a0 − a)2 .

We have a0 − a ∈ m. Together with the above equality, this implies that a0 − a ∈ So we have a0 − a = 0.

∞ \

mi .

i=1

Let A → A0 be a local homomorphism of local rings. We say A0 is essentially etale over A if there exists an etale A-algebra B and a prime ideal p of B lying above the maximal ideal of A such that A0 is A-isomorphic to Bp . Using the fact that Spec B → Spec A is quasi-finite, one can show the maximal ideal of A0 is the only prime ideal of A0 lying above the maximal

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ideal of A. So if A0 and A00 are two local essentially etale A-algebras, then any A-homomorphism A0 → A00 is local. A local homomorphism A → A0 of locally rings is called strictly essentially etale if it is essentially etale and the homomorphism A → A0 induces an isomorphism on the residue field. Using 1.10.9 and 2.3.7, one can show if A → A0 and A0 → A00 are essentially etale local homomorphisms, then their composite A → A00 is also essentially etale. Lemma 2.8.5. Let A and A0 be local rings, A → A0 a strictly essentially etale local homomorphism, and (R, m) a henselian local ring. Then the canonical map loc.Hom(A0 , R) → loc.Hom(A, R) is bijective, where loc.Hom denotes the set of local homomorphisms. Proof. Given a local homomorphism A → R, we need to show it can be extended uniquely to a local homomorphism A0 → R. Consider Spec R as a scheme over Spec A. We need to show that there exists a unique Amorphism Spec R → Spec A0 that maps the closed point of Spec R to the closed point of Spec A0 , or equivalently, that there exists a unique section for the projection Spec (A0 ⊗A R) → Spec R

that maps the closed point of Spec R to the point in Spec (A0 ⊗A R) that is above the closed points in Spec A0 and Spec R. Note that since A → A0 induces an isomorphism of residue fields, there is one and only one point in Spec (A0 ⊗A R) that is above the closed points in Spec A0 and Spec R. Let B be an etale A-algebra such that A0 is A-isomorphic to Bp for some prime ideal p of B. It suffices to show there exists a unique section for the projection Spec (B ⊗A R) → Spec R that maps the closed point of Spec R to the unique point in Spec (B ⊗A R) that is above the closed point in Spec R and the point p in Spec B. This point has the same residue field as R and hence is the image of a unique section for the projection Spec (B ⊗A R/m) → Spec R/m. By 2.8.3 (vii), this section is induced by a section of the projection Spec (B ⊗A R) → Spec R, which has the required property. The uniqueness follows from 2.3.10 (i).

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Lemma 2.8.6. Let A be a local ring, and let A1 and A2 be strictly essentially etale local A-algebras. (i) There exists a strictly essentially etale local A-algebra A3 dominating A1 and A2 , that is, there exist A-homomorphisms (necessarily local) from Ai (i = 1, 2) to A3 . (ii) There exists at most one A-homomorphism from A1 to A2 . Proof. (i) We can find etale A-algebras Bi (i = 1, 2) and prime ideals pi of Bi such that Ai are A-isomorphic to (Bi )pi , respectively. There exists a unique prime ideal p of B1 ⊗A B2 lying above p1 and p2 . We can take A3 = (B1 ⊗A B2 )p . (ii) Keep the above notation. It suffices to show that there exists at most one A-morphism from Spec A2 to Spec B1 that maps the closed point of Spec A2 to p1 , or equivalently, that there exists at most one section for the projection Spec (A2 ⊗A B1 ) → Spec A2 that maps the closed point of Spec A2 to the (unique) point of Spec (A2 ⊗A B1 ) that is above p1 ∈ Spec B1 and the closed point of Spec A2 . This follows from 2.3.10 (i). Let A be a local ring. A local homomorphism A → A0 is called essentially of finite type if there exists a finitely generated A-algebra B such that A0 is A-isomorphic to Bp for some prime ideal p of B lying above the maximal ideal of A. There exists a set of local A-algebras essentially of finite type such that every local A-algebra essentially of finite type is isomorphic to a member of this set. For example, we can take this set to be the family of A-algebras of the form (A[t1 , . . . , tn ]/I)p , where n goes over the set of nonnegative integers, I goes over the set of ideals of A[t1 , . . . , tn ], and p goes over the set of prime ideals of A[t1 , . . . , tn ]/I lying above the maximal ideal of A. Therefore there exists a set of strictly essentially etale local A-algebras such that any strictly essentially etale local A-algebra is isomorphic to a member of this set. Let S be the category whose objects are strictly essentially etale local A-algebras and whose morphisms are Ahomomorphisms. By 2.8.6, S satisfies the condition (I2) and (I3) in 2.7. We define the henselization Ah of A to be Ah =

lim −→

A0 ∈ob S

A0 .

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b = Lemma 2.8.7. Let A be a ring, I a finitely generated ideal of A, A n n b limn A/I , and Jn = ker (A → A/I ). Then ←− b induces isomorphisms Jn = (i) The canonical homomorphism A → A n b n ∼ b nb n n+1 ∼ n b n+1 b I A, A/I = A/I A, and I /I A. = I A/I b is noetherian. (ii) If A/I is noetherian, then A

Proof. b → A/I n is surjective. Its kernel is (i) It is clear that the projection A n ∼ b Jn . It follows that A/Jn = A/I . Consider the commutative diagram 0 → I n /I m+n → A/I m+n → A/I n → 0 ↓ ↓ ↓ b m+n → A/J b n →0 0 → Jn /Jm+n → A/J ∼ ∼ ↓ = ↓ =↓ n m+n m+n 0 → I /I → A/I → A/I n → 0,

where the first three vertical arrows are induced by the canonical hob and the last three vertical arrows are inmomorphism i : A → A, b duced by the projection A → A/I m+n . By the five lemma, the vertical arrowJn/Jm+n → I n /I m+n is an isomorphism. It follows that the vertical arrow I n /I m+n → Jn /Jm+n is an isomorphism. So we have Jn = i(I n ) + Jm+n . Let x1 , . . . , xk ∈ I be a finite family of generators of I. For any z ∈ Jn , since Jn = i(I n ) + Jn+1 , we have X (0) z=i ai1 ...ik xi11 · · · xikk + zn+1 i1 +···+ik =n

(0)

for some ai1 ...ik ∈ A and zn+1 ∈ Jn+1 . Since Jn+1 = i(I n+1 ) + Jn+2 , we have X (1) zn+1 = i ai1 ...ik xi11 · · · xikk + zn+2 i1 +···+ik =n

(1)

for some ai1 ...ik ∈ I and zn+2 ∈ Jn+2 . Repeating this process, we get a (m) ai1 ...ik

family such that

z=i

∈ I m and zn+m+1 ∈ Jn+m+1 (i1 + · · · + ik = n, m = 0, 1, . . .) X

(0) (m) (ai1 ...ik + · · · + ai1 ...ik )xi11 · · · xikk + zn+m+1

i1 +···+ik =n

for any m ≥ 0. Taking limits on both sides, we get X (0) (1) z= (i(ai1 ...ik ) + i(ai1 ...ik ) + · · · )i(xi11 · · · xikk ). i1 +···+ik =n

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b Hence Jn ⊂ I n A. b It is clear that I n A b ⊂ So we have z ∈ I n A. n b n m+n ∼ Jn . So we have Jn = I A. Since I /I = Jn /Jm+n , we have b m+n A. b In particular, we have A/I n ∼ b nA b and I n /I m+n ∼ = I n A/I = A/I n n+1 ∼ n b n+1 b I /I A. = I A/I (ii) By [Atiyah and Macdonald (1969)] 10.25, it suffices to show that L∞ L n b n+1 b n n+1 A) is a noetherian ring, that is, ∞ is a noethen=0 (I A/I n=0 I /I rian ring. This follows from [Atiyah and Macdonald (1969)] 10.7. Proposition 2.8.8. Let (Aλ , φλµ ) be a direct system of local rings with φλµ being local homomorphisms. For each λ, let mλ be the maximal ideal of Aλ and let kλ = Aλ /mλ be the residue field. (i) A = limλ Aλ is a local ring with the maximal ideal m = limλ mλ , and −→ −→ its residue field A/m is isomorphic to limλ kλ . −→ (ii) If mµ = mλ Aµ for any pair µ ≥ λ, then m = mλ A for any λ. (iii) If Aµ is flat over Aλ for any pair µ ≥ λ, then A is flat over Aλ for any λ. (iv) If mµ = mλ Aµ and Aµ is flat over Aλ for any pair µ ≥ λ, and if Aλ is noetherian for any λ, then A is noetherian. Proof.

Note that m = limλ mλ is an ideal of A, and we have −→ A/m ∼ A /m = lim k , = lim −→ λ λ −→ λ λ

λ

which is a field. It follows that m is a maximal ideal of A. We have A − m = lim(Aλ − mλ ). −→ λ

As Aλ − mλ are the groups of units of Aλ , A − m is the group of units of A. So A is a local ring with the maximal ideal m. If mµ = mλ Aµ for any pair µ ≥ λ, then we have m = lim mλ Aµ = mλ A −→ µ≥λ

for any λ. If Aµ is flat over Aλ for any pair µ ≥ λ, then A ∼ A = lim −→µ≥λ µ is flat over Aλ for any λ. Suppose that all the above conditions hold and suppose furthermore that each Aλ is noetherian. Then each mλ is finitely b = lim A/mn is generated. So m = mλ A is finitely generated. By 2.8.7, A ←−n b∼ b nA noetherian, and A/m = A/mn . Since A/mn = A/mnλ A ∼ = Aλ /mnλ ⊗Aλ A

b is flat over b nA and A is flat over Aλ , A/mn is flat over Aλ /mnλ . So A/m n b Aλ /mλ . By 1.3.5 (iv), A is flat over Aλ . Let I be a finitely generated ideal

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of A. Then I = Iλ A for some ideal Iλ of Aλ . Since A is flat over Aλ , we b is flat over Aλ , the canonical homomorphism have Iλ ⊗Aλ A ∼ = I. Since A b→A b is injective, that is, Iλ ⊗Aλ A b→A b Iλ ⊗Aλ A ⊗A A

b→A b is injective. By is injective. So the canonical homomorphism I ⊗A A b b 1.1.1 (ix), A is flat over A. Note that A is local and the canonical homob is local. Indeed, A/m b A b is a field since it is isomorphic morphism A → A b − mA, b the image of x under each projection to A/mA. For any x ∈ A b is faithfully flat limn A/mn → A/mn is a unit, and hence x is a unit. So A ←− over A. If I1 ⊂ I2 ⊂ · · ·

is an ascending chain of ideals of A, then the chain b ⊂ I2 A b ⊂ ··· I1 A

b is noetherian. But Ii A b ∩ A = Ii for all i by 1.2.6 (v). is stationary since A So we have Ii0 = Ii0 +1 = · · ·

for a large i0 . Hence A is noetherian.

Proposition 2.8.9. Let (A, m) be a local ring. (i) Ah is a henselian local ring, the canonical homomorphism A → Ah is local and faithfully flat, mAh is the maximal ideal of Ah , and the homomorphism A/m → Ah /mAh is an isomorphism. (ii) For any henselian local ring R, the canonical map loc.Hom(Ah , R) → loc.Hom(A, R) is bijective. (iii) If A is henselian, then A ∼ = Ah . ch is an isomorphism. b→A (iv) The canonical homomorphism A h (v) If A is noetherian, then so is A . Proof. (i) For every strictly essentially etale local A-algebra A0 , the homomorphism A → A0 is local and faithfully flat, mA0 is the maximal ideal of A0 , and A/m → A0 /mA0 is an isomorphism. By 2.8.8, Ah is a local ring, the canonical homomorphism A → Ah is local and faithfully flat, mAh is the maximal ideal of Ah , and the homomorphism A/m → Ah /mAh is an isomorphism.

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Let us verify that the condition 2.8.3 (vii) holds for Ah . Given an etale morphism g : X → Spec Ah and a section s : Spec Ah /mAh → X ⊗Ah Ah /mAh of the fiber of g over the closed point of Spec Ah , let us prove that this section can be lifted to a section of g. Shrinking X, we may assume that g has finite presentation. By 1.10.9 and 2.3.7, we can find a strictly essentially etale local A-algebra A0 and an etale morphism g 0 : X 0 → Spec A0 inducing g after base change. Let x0 ∈ X 0 be the image of the composite s

Spec Ah /mAh → X ⊗Ah Ah /mAh → X → X 0 . Then OX 0 ,x0 is a strictly essentially etale local A-algebra. So we have an A0 homomorphism OX 0 ,x0 → Ah . It induces an A0 -morphism Spec Ah → X 0 . The graph of this morphism is a section of g extending s. (ii) follows from 2.8.5. (iii) By 2.8.3 (vii), if A is henselian, then any strictly essentially etale local A-algebra is isomorphic to A. So A ∼ = Ah . h (iv) Since A is flat over A, we have mn Ah /mn+1 Ah ∼ = mn /mn+1 ⊗A Ah ∼ = mn /mn+1 ⊗A/m Ah /mAh ∼ = mn /mn+1 ⊗A/m A/m ∼ = mn /mn+1

for all n. By induction on n, this implies that A/mn ∼ = Ah /mn Ah ch . b∼ for all n. So A =A (v) follows from 2.8.8.

Proposition 2.8.10. Let A be a local noetherian ring. Then A is reduced (resp. regular, resp. normal) if and only if Ah is so. Proof.

Use the same argument as the proof of 2.4.4.

Proposition 2.8.11. Let (Rλ , φλµ ) be a direct system of local rings such that φλµ are local homomorphisms. (i) If each Rλ is henselian, so is limλ Rλ . −→ (ii) In general, we have ∼ lim Rh . (lim Rλ )h = −→ −→ λ λ

λ

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Proof. (i) For each λ, let mλ be the maximal ideal of Rλ , let R = limλ Rλ and −→ let m = limλ mλ . Then R is a local ring with the maximal ideal m. Let −→ us check that the condition 2.8.3 (iv) holds. Let A = R[t]/(f (t)), where f (t) ∈ R[t] is a monic polynomial. We need to show that the canonical homomorphism A → A/mA induces a one-to-one correspondence on the sets of idempotent elements in A and in A/mA. By the proof of 2.8.3 (iv)⇒(iii), A → A/mA is injective when restricted to the set of idempotent elements in A. Let e¯ be an idempotent element in A/mA. We can find λ such that there exists a monic polynomial fλ (t) ∈ Rλ [t] whose image in R[t] is f (t) and an idempotent element e¯λ ∈ Aλ /mλ Aλ whose image in A/mA is e¯, where Aλ = Rλ [t]/(fλ (t)). Since Rλ is henselian, there exists an idempotent element eλ in Aλ lifting e¯λ . Let e be the image of eλ in A. Then e is idempotent element lifting e¯. (ii) By 2.8.9 (ii), the canonical homomorphisms Rλ → limλ Rλ induce −→ homomorphisms Rλh → (limλ Rλ )h and hence a homomorphism −→ Φ : lim Rλh → (lim Rλ )h . −→ −→ λ

λ

lim Rh −→λ λ

By (i), is henselian. By 2.8.9 (ii), the canonical homomorphism limλ Rλ → limλ Rλh induce a homomorphism −→ −→ Ψ : (lim Rλ )h → lim Rλh . −→ −→ λ

λ

One can verify that Φ and Ψ are inverse to each other.

Proposition 2.8.12. Let (A, m) be a local ring, B a finite A-algebra, and n1 , . . . , nk all the maximal ideals of B. Then we have B ⊗A Ah ∼ = (Bn1 )h × · · · × (Bnk )h . Proof. Since A → Ah induces an isomorphism on residue fields, the fibers of Spec B → Spec A and Spec (B ⊗A Ah ) → Spec Ah above the closed points of Spec A and Spec Ah are isomorphic. Since B (resp. B ⊗A Ah ) is finite over A (resp. Ah ), the fiber of Spec B → Spec A (resp. Spec (B ⊗A Ah ) → Spec Ah ) above the closed point consists of maximal ideals of B (resp. B ⊗A Ah ). So Spec (B ⊗A Ah ) → Spec B induces a one-to-one correspondence between the set of maximal ideals of B ⊗A Ah and the set of maximal ideals of B. Let n01 , . . . , n0k be the maximal ideals of B ⊗A Ah lying above n1 , . . . , nk , respectively. By 2.8.3 (i), we have B ⊗A Ah ∼ = (B ⊗A Ah )n01 × · · · × (B ⊗A Ah )n0k .

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To prove our assertion, it suffices to show (B ⊗A Ah )n0i ∼ = (Bni )h for all h i. For each i, let ei be the element in B ⊗A A so that its projection to (B ⊗A Ah )n0j is 0 if j 6= i and 1 if j = i. We have e2i = ei ,

ei ej = 0 (i 6= j),

e1 + · · · + ek = 1.

Since B ⊗A Ah = limA0 ∈ob S B ⊗A A0 , there exists a strictly essentially −→ etale local A-algebra A0 such that e1 , . . . , ek are images of some elements e01 , . . . , e0k ∈ B ⊗A A0 with the property 0 e02 i = ei ,

e0i e0j = 0

(i 6= j),

e01 + · · · + e0k = 1.

We then have B ⊗A A0 ∼ = (B ⊗A A0 )e01 × · · · × (B ⊗A A0 )e0k . On the other hand, we have (B ⊗A A0 )e0i ⊗A0 Ah ∼ = (B ⊗A Ah )ei ∼ = (B ⊗A Ah )n0i . As Spec (B ⊗A A0 )e0i → Spec A0 and Spec (B ⊗A A0 )e0i ⊗A0 Ah → Spec Ah have the same fiber over the closed point of Spec A0 and Spec Ah , it follows that (B ⊗A A0 )e0i is local and its maximal ideal lies above ni . Moreover (B ⊗A A0 )e0i is strictly essentially etale over Bni , so we have (Bni )h ∼ = ((B ⊗A A0 )e0i )h . We also have Ah ∼ = A0h . Replacing A by A0 and B by B ⊗A A0 , we are reduced to the case where B is a product of local rings. It suffices to treat the case where B is local. In this case, B ⊗A Ah is local. One can check that it satisfies the condition 2.8.3 (i). So it is henselian. Moreover, we have (B ⊗A A0 ) and each B ⊗A A0 is a strictly essentially B ⊗A Ah ∼ = lim −→A0 ∈ob S etale local B-algebra. So we have B ⊗A Ah ∼ = Bh. Proposition 2.8.13. Let R be a henselian local ring, S = Spec R, and s the closed point of S. For any smooth morphism g : X → S, any section of gs : Xs = X ⊗R k(s) → Spec k(s) can be lifted to a section of g. Proof. Let hs : Spec k(s) → Xs be a section of gs , and let x be the image of hs . Since g is smooth, there exists an open neighborhood U of x admitting an etale S-morphism j : U → AnS = Spec R[t1 , . . . , tn ].

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Let y = j(x). We have k(x) ∼ = k(y) ∼ = k(s). So OX,x and OAnS ,y have the same henselization. Denote the common henselization by A. Consider the commutative diagram Spec A ↓& j

U → AnS . g|U ↓ . S It suffices to construct a section for Spec A → S because such a section composed with the canonical morphism Spec A → X gives rise to a section of g inducing hs . Since k(y) ∼ ¯1 , . . . , a ¯n ∈ k(s) such that = k(s), there exist a the image of y in Ans = Spec k(s)[t1 , . . . , tn ] corresponds to the maximal ideal (t1 − a ¯1 , . . . , tn − a ¯n ) of k(s)[t1 , . . . , tn ]. Let ai ∈ R (i = 1, . . . n) be liftings of a ¯i , respectively. Then y corresponds to the maximal ideal of R[t1 , . . . , tn ] generated by the maximal ideal of R and t1 − a1 , . . . , tn − an . The R-homomorphism R[t1 , . . . , tn ] → R,

ti 7→ ai

induces an R-homomorphism A → R by passing to localization and then to henselization. The corresponding morphism Spec R → Spec A is a section for Spec A → S. Proposition 2.8.14. Let (R, m) be a local ring, k = R/m the residue field, S = Spec R, and s the closed point of S. The following conditions are equivalent: (i) R is strictly henselian. (ii) R is henselian, and any finite etale S-scheme X is isomorphic to a disjoint union of copies of S. (iii) For any etale morphism g : X → S and any point x ∈ X lying above s, there exists a section h : S → X of g such that h(s) = x. Proof. (i)⇒(iii) The residue field of X at x is isomorphic to k since it is finite separable over k and k is separably closed. So there exists a section of gs : X ⊗R k → Spec k with image x. We then apply 2.8.3 (vii). (iii)⇒(ii) By 2.8.3, R is henselian. Suppose that X = Spec B for some finite etale R-algebra B. Then B is a direct product of finitely many local rings. Without loss of generality, assume B local. The closed point of X is above the closed point of S. If the condition (iii) holds, then X → S admits a section. Since X and S are connected, we must have X ∼ = S by 2.3.10 (i).

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(ii)⇒(i) We need to show that k is separably closed. If this is not true, then there exists a monic irreducible polynomial f¯(t) ∈ k[t] of degree > 1 such that f¯(t) and f¯0 (t) generate the ideal k[t]. Let f (t) ∈ R[t] be a monic polynomial lifting f¯(t). Applying Nakayama’s lemma to R[t]/(f (t)), we see f (t) and f 0 (t) generate the ideal R[t]. By 2.3.3, R[t]/(f (t)) is etale and finite over R. If the condition (ii) holds, then R[t]/(f (t)) is isomorphic to a direct product of copies of R as an R-algebra. This implies that f (t) is a product of linear polynomials. Then f¯(t) has the same property. This contradicts the fact that f¯(t) is irreducible and has degree > 1. We leave the proof of 2.8.15–19 below to the reader. Lemma 2.8.15. Let A be a local ring, A0 an essentially etale local Aalgebra, R a henselian local ring, and k(A), k(A0 ) and k(R) the residue fields of A, A0 and R, respectively. Given a local homomorphism φ : A → R and a k(A)-homomorphism α : k(A0 ) → k(R), there exists a unique local A-homomorphism φ0 : A0 → R inducing the homomorphism α on residue fields. Lemma 2.8.16. Let A be a local ring, let A1 and A2 be essentially etale local A-algebras, let k(A), k(A1 ) and k(A2 ) be the residue fields of A, A1 and A2 respectively, and let k be a field containing k(A). (i) For k(A)-homomorphisms βi : k(Ai ) → k (i = 1, 2), there exist an essentially etale local A-algebra A3 and a k(A)-homomorphism β3 : k(A3 ) → k such that (A3 , β3 ) dominates (Ai , βi ) (i = 1, 2), that is, there exist A-homomorphisms (necessarily local) φi : Ai → A3 such that βi = β3 φ¯i , where φ¯i are the homomorphisms induced by φi on residue fields. (ii) For any k(A)-homomorphism γ : k(A1 ) → k(A2 ), there exists at most one A-homomorphism from A1 to A2 inducing the homomorphism γ on residue fields. Let (A, m) be a local ring, k = A/m its residue fields, Ω a separable closure of k, and i : k → Ω the inclusion. Let T be the category defined as follows: Objects in T are pairs (A0 , βA0 ), where A0 are essentially etale local A-algebras and βA0 : k(A0 ) → Ω are k-homomorphisms from the residue fields k(A0 ) of A0 to Ω. A morphism (A1 , βA1 ) → (A2 , βA2 ) in T is an A¯ where φ¯ is the algebra homomorphism φ : A1 → A2 such that βA1 = βA2 φ, homomorphism induced by φ on residue fields. By 2.8.16, T satisfies the condition (I2) and (I3) in 2.7. There exists a full subcategory of T whose objects form a set such that any object in T is isomorphic to an object in

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this subcategory. We define the strict henselization Ahs i of A relative to i to be Ahs i = (A0 ,β

lim −→

A0 .

A0 )∈ob T

We also call Ahs i the strict localization of A relative to i. We often denote hs e Ahs by A or A. i

Proposition 2.8.17. Let (A, m) be a local ring, k = A/m the residue field, Ω a separable closure of k, and i : k → Ω the inclusion. (i) Ahs i is a strictly henselian local ring, the canonical homomorphism hs hs A → Ahs i is local and faithfully flat, mAi is the maximal ideal of Ai , and hs the residue field of Ai is k-isomorphic to Ω. (ii) Let R be a henselian local ring with residue field k(R), φ : A → R a local homomorphism, and α : Ω → k(R) a homomorphism such that αi : k → k(R) coincides with the homomorphism induced by φ. Then there exists a unique local A-homomorphism φ0 : Ahs → R inducing the i homomorphism α on residue fields. (iii) If A is strictly henselian, then A ∼ = Ahs i . hs (iv) If A is noetherian, then so is Ai . (v) Suppose i0 : k → Ω0 is the inclusion of k into another separable closure of k. Then for any k-isomorphism σ : Ω → Ω0 , the A-homomorphism hs Ahs i → Ai0 defined in (ii) is an isomorphism. We have ∼ Aut(Ahs i /A) = Gal(Ω/k).

Proposition 2.8.18. Let A be a local noetherian ring. Then A is reduced (resp. regular, resp. normal) if and only if Ahs is so. Proposition 2.8.19. Let (Rλ , φλµ ) be a direct system of local rings such that φλµ are local homomorphisms. (i) If each Rλ is strictly henselian, so is limλ Rλ . −→ (ii) In general, let i be the inclusion of the residue field of limλ Rλ in −→ one of its separable closure Ω, and for each λ, let iλ be the inclusion of the residue field of Rλ in its separable closure in Ω. Then we have ∼ (lim Rλ )hs (R )hs . i = lim −→ −→ λ iλ λ

λ

˜a Proposition 2.8.20. Let (A, m) be a local ring, B a finite A-algebra, n hs ˜ in B. Then n is maximal ideal of B ⊗A A , and n the inverse image of n a maximal ideal of B, the residue field k(˜ n) = (B ⊗A Ahs )/˜ n is a separable

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closure of the residue field k(n) = B/n, and (B ⊗A Ahs )n˜ is isomorphic to the strict henselization of Bn with respect to the separable closure k(˜ n) of k(n). ˜ lies above the maximal ideal Proof. Since B ⊗A Ahs is finite over Ahs , n hs of A , and hence above the maximal ideal of A. So n is above the maximal ideal of A. As B is finite over A, n is a maximal ideal of B. Note that k(˜ n) is isomorphic to the residue field of k(n) ⊗k(A) k(Ahs ) at a prime ideal, where k(A) and k(Ahs ) are the residue fields of A and Ahs , respectively. Since k(n) is finite over k(A) and k(Ahs ) is a separable closure of k(A), k(˜ n) is a separable closure of k(n). Since Ahs is henselian and B ⊗A Ahs is finite over Ahs , we have B ⊗A Ahs ∼ = (B ⊗A Ahs )n˜ × · · · × (B ⊗A Ahs )n˜ , 1

k

˜i (i = 1, . . . , k) are all the maximal ideals of B ⊗A Ahs . For each i, where n let ei be the element in B ⊗A Ahs so that its projection to (B ⊗A Ahs )n˜j is 0 if j 6= i and 1 if j = i. We have e2i = ei , hs

ei ej = 0 (i 6= j),

e1 + · · · + ek = 1.

e0i e0j = 0

e01 + · · · + e0k = 1.

0

Since B ⊗A A = limA0 ∈ob T B ⊗A A , there exists an essentially etale −→ local A-algebra A0 in T such that e1 , . . . , ek are images of some elements e01 , . . . , e0k ∈ B ⊗A A0 with the property 0 e02 i = ei ,

We then have

(i 6= j),

B ⊗A A0 ∼ = (B ⊗A A0 )e01 × · · · × (B ⊗A A0 )e0k . On the other hand, we have (B ⊗A A0 )e0i ⊗A0 Ahs ∼ = (B ⊗A Ahs )ei ∼ = (B ⊗A Ahs )n˜i .

In particular, (B ⊗A A0 )e0i ⊗A0 Ahs is local. Note that the canonical morphism Spec ((B ⊗A A0 )e0i ⊗A0 Ahs )⊗Ahs Ahs /mAhs → Spec (B ⊗A A0 )e0i ⊗A0 A0 /mA0

is surjective. It follows that (B ⊗A A0 )e0i is local and its maximal ideal ˜i ∩ B of B. Moreover (B ⊗A A0 )e0i is lies above the maximal ideal ni = n essentially etale over Bni , so we have (Bn )hs ∼ = ((B ⊗A A0 )e0 )hs . i

i

We also have Ahs ∼ = A0hs . Replacing A by A0 and B by (B ⊗A A0 )e0i , we are reduced to the case where B and B⊗A Ahs are local. In this case, B⊗A Ahs is (B⊗A A0 ) and a strictly henselian local ring. We have B⊗A Ahs ∼ = lim −→A0 ∈ob T each B ⊗A A0 is an essentially etale local B-algebra. So we have B ⊗A Ahs ∼ = B hs .

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([SGA 1] I 10, [EGA IV] 18.10.) Proposition 2.9.1. Let f : X → Y be a morphism locally of finite type between noetherian schemes. Assume that for any y ∈ f (X), OY,y is normal. The following conditions are equivalent: (i) f is etale. (ii) f is unramified and the canonical homomorphism OY,f (x) → OX,x is injective for any x ∈ X. (iii) f is unramified and dim OY,f (x) ≤ dim OX,x for any x ∈ X. Proof. (i)⇒(iii) is clear. (iii)⇒(ii) Let I be the kernel of the homomorphism OY,f (x) → OX,x . Shrinking Y , we may assume I = If (x) for some coherent ideal I of OY . Let Y1 be the closed subscheme of Y with the ideal sheaf I . Shrinking X and Y , we may assume that f can be factorized as a composite X → Y1 → Y. We have dim OX,x ≤ dim OY1 ,f (x) + dim OX,x /mY1 ,f (x) OX,x by [Matsumura (1970)] (13.B) Theorem 19 (1). Note that X → Y1 is unramified and hence dim OX,x /mY1 ,f (x) OX,x = 0. So we have dim OX,x ≤ dim OY1 ,f (x) , and hence dim OY,f (x) ≤ dim OY1 ,f (x) . Since OY,f (x) is normal and hence an integral domain, this implies that I = 0. So OY,f (x) → OX,x is injective. (ii)⇒(i) For any x ∈ X, let OeY,f (x) be the strict henselization of OY,f (x) e = X ×Y Spec OeY,f (x) , with respect to a separable closure of k(f (x)), let X 0 e and let x ∈ X be a point lying above x ∈ X and the closed point of Spec OeY,f (x) . We claim that the homomorphism OeY,f (x) → OX,x e 0 is injective. Indeed, OY,f (x) is normal and hence an integral domain. Let y1 ∈ Y be the point corresponding to the generic point of Spec OY,f (x) . Since OY,f (x) → OX,x is injective, the ring OX,x ⊗OY,f (x) OY,y1 is nonzero. Hence there exists x1 ∈ X lying above y1 such that x ∈ {x1 }. Since OX,x e 0 e lying above x1 such that is faithfully flat over OX,x , there exists x01 ∈ X x0 ∈ {x01 }. By 2.8.18, OeY,f (x) is normal and hence an integral domain. Since the generic point of Spec OeY,f (x) is the only point lying above the generic

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point in Spec OY,f (x) , and x01 lies above the generic point of Spec OY,f (x) , it follows that x01 lies above the generic point of Spec OeY,f (x) . This implies that the homomorphism OeY,f (x) → OX,x e 0 is injective. e e Note that X → Spec OY,f (x) is unramified and hence quasi-finite. Let U e By the Zariski Main Theorem be an affine open neighborhood of x0 in X. 1.10.13, U → Spec OY,f (x) can be factorized as

j U ,→ U → Spec OeY,f (x)

such that j is an open immersion and U → Spec OeY,f (x) is finite. By 2.8.3 (i), we have a U= Spec OU ,¯x , x ¯

where x¯ goes over the set of points of U above the closed point of Spec OeY,f (x) . Note that j(x0 ) is such a point. Since j is an open immersion, we have ∼ ∼ OX,x e 0 = OU,x0 = OU ,j(x0 ) .

Since U → Spec OeY,f (x) is finite, OU ,j(x0 ) is finite over OeY,f (x) . So OX,x e 0 e e is finite over OY,f (x) . Let m be the maximal ideal of OY,f (x) . Since e → Spec OeY,f (x) is unramified and the residue field of OeY,f (x) is sepaX

rably closed, we have

OeY,f (x) /m ∼ = OX,x e 0 /mOX,x e 0.

By Nakayama’s lemma, OeY,f (x) → OX,x e 0 is surjective. We have shown it is injective. So we have OeY,f (x) ∼ = OX,x e 0.

Consider the commutative diagram

OY,f (x) → OX,x ↓ ↓ ∼ = e OY,f (x) → OX,x e 0.

Since OeY,f (x) is faithfully flat over OY,f (x) , OX,x e 0 is faithfully flat over OY,f (x) . On the other hand, OX,x e 0 is faithfully flat over OX,x . It follows that OX,x is faithfully flat over OY,f (x) . This is true for all x ∈ X. So f : X → Y is flat. Since it is unramified, it is etale.

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Proposition 2.9.2. Let Y be a normal connected noetherian scheme and let X be an etale separated Y -scheme of finite type. Then each connected component Xi of X is normal and is an open subscheme of the normalization of Y in the function field of Xi . Xi is finite over Y if and only if it is equal to the this normalization. Proof. Let Ki and K be the function fields of Xi and Y , respectively, and let Yi0 be the normalization of Y in Ki . Note that Yi0 is finite over Y since Ki is finite separable over K. By 2.4.4, X is normal. So we have a birational Y -morphism Xi → Yi0 . Since Xi is quasi-finite and separated over Y , Xi → Yi0 is also quasi-finite and separated. By the Zariski Main Theorem 1.10.13, it can be factorized as a composite Xi ,→ Xi0 → Yi0 such that Xi ,→ Xi0 is an open immersion, and Xi0 → Yi0 is finite. Replacing Xi0 by the closure of Xi with the reduced closed subscheme structure, we may assume that Xi0 is integral. Note that Xi0 → Yi0 is a finite birational morphism. Since Yi0 is normal, we have Xi0 ∼ = Yi0 . So Xi → Yi0 is an open immersion. If Xi is finite over Y , then Xi is finite over Yi0 and hence Xi ∼ = Yi0 . Let Y be a normal connected noetherian scheme, K its function field, and L a finite separable extension of K. We say L is unramified over Y if the normalization of Y in L is unramified over Y , or equivalently, etale over Y . By 2.9.2, the category of connected finite etale Y -schemes is equivalent to the category of finite separable extensions of K(Y ) unramified over Y . Suppose L = L1 × · · · × Ln for some finite separable extensions Li of K. We say that L is unramified over Y if each Li is unramified over Y . Proposition 2.9.3. Let Y be a normal connected noetherian scheme and let K be its function field. (i) K is unramified over Y . (ii) Let L be an extension of K unramified over Y , M an extension of L unramified over the normalization of Y in L. Then M is unramified over Y. (iii) Let Y 0 be a normal connected noetherian Y -scheme dominating Y and let K 0 be its function field. If L is an extension of K unramified over Y , then L ⊗K K 0 is unramified over Y 0 . If Y = Spec A and Y 0 = Spec A0 are affine and B is the integral closure of A in L, then B ⊗A A0 is the integral closure of A0 in L ⊗K K 0 .

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(iv) If L1 and L2 are extensions of K unramified over Y , then L1 ⊗K L2 is unramified over Y . Proof. (i) is obvious. (ii) follows from 2.2.2 (iii). (iii) It suffices to consider the case where Y = Spec A and Y 0 = Spec A0 are affine. Note that Spec (B ⊗A A0 ) is finite and etale over Spec A0 . So Spec (B ⊗A A0 ) is normal. It function ring is isomorphic to B ⊗A A0 ⊗A0 K 0 ∼ = B ⊗A K ⊗K K 0 ∼ = L ⊗K K 0 .

So B ⊗A A0 is the integral closure of A0 in L ⊗K K 0 , and L ⊗K K 0 is unramified over Spec A0 . (iv) follows from (ii) and (iii).

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Etale Fundamental Groups

3.1

Finite Group Actions on Schemes

([SGA 1] V 1.) Let X be a scheme on which a finite group G acts on the right. If X = Spec A, this is equivalent to saying G acts on A on the left. For any scheme Z, G acts on the left on the set Hom(X, Z). Let Hom(X, Z)G be the subset of Hom(X, Z) consisting of morphisms invariant under G. A morphism X → Y invariant under the action of G is called the quotient of X by G if the canonical map Hom(Y, Z) → Hom(X, Z)G is bijective for any scheme Z, that is, Y represents the functor Z 7→ Hom(X, Z)G . We often denote Y by X/G. Proposition 3.1.1. Let A be a ring on which a finite group G acts on the left, B = AG , X = Spec A, Y = Spec B, and π : X → Y the morphism corresponding to the homomorphism AG ,→ A. (i) X is integral over Y . (ii) π is surjective. Its fibers are orbits of G, and the topology on Y is the quotient topology induced from X. (iii) Given x ∈ X, let y = π(x) and let Gx = {g ∈ G|gx = x} be the stabilizer of x. Then the residue field k(x) is a normal algebraic extension of the residue field k(y), and the canonical homomorphism Gx → Gal(k(x)/k(y)) is surjective. (iv) The canonical morphism OY → (π∗ OX )G is an isomorphism, and Y is the quotient of X by G. 117

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Proof. Q (i) For any a ∈ A, the polynomial g∈G (t − ga) in A[t] is invariant under the action of G. So it lies in B[t]. a is a root of this polynomial. So A is integral over B. (ii) By [Atiyah and Macdonald (1969)] 5.10, π is surjective, and π(V (I)) = V (I ∩ B) for any ideal I of A. So π is a closed map, and hence the topology on Y is the quotient topology induced from X. Let q be a prime ideal of B, and let p and p0 be prime ideals of A such that q = p ∩ B = p0 ∩ B. Then for any a ∈ p0 , we have Y g∈G

ga ∈ p0 ∩ B = q ⊂ p.

So ga ∈ p for some g ∈ G. It follows that p0 ⊂ ∪g∈G gp. This implies that p0 ⊂ gp for some g ∈ G by [Matsumura (1970)] (1.B). We have gp ∩ B = g(p ∩ B) = p ∩ B = p0 ∩ B.

By [Atiyah and Macdonald (1969)] 5.9, we have p0 = gp. So the fibers of π are orbits of G. (iii) Let p be the prime ideal of A corresponding to x and let q = p ∩ B be the prime ideal corresponding to y. Replacing A and B by A ⊗B Bq and Bq , respectively, we may assume B is a local ring with the maximal ideal q. Then p is a maximal ideal of A. Any a ∈ A is a root of the polynomial Q of g∈G (t − ga) ∈ B[t]. So every element in k(x) is a root of a polynomial in k(y)[t] splitting in k(x) with degree |G|. Hence k(x)/k(y) is a normal algebraic extension. Since any finite separable extension is generated by one element, the separable closure k(y)s of k(y) in k(x) has degree ≤ |G|. Let a ∈ A so that its image a ¯ in k(x) lies in k(y)s and generates k(y)s over k(y). For any g ∈ G − Gx , gp is a maximal ideal of A distinct from p. By the Chinese remainder theorem, there exists a0 ∈ A such that a0 − a ∈ p and a0 ∈ gp for any g 6∈ G − Gx . Let ga0 be the image of ga0 in k(x), and let Y f (t) = (t − ga0 ). g∈G

We have f (t) ∈ k(y)[t] and f (a0 ) = 0. Given τ ∈ Gal(k(x)/k(y)), τ (a0 ) is a root of f (t). So τ (a0 ) = ga0 for some g ∈ G. If g 6∈ Gx , we have ga0 ∈ p and hence ga0 = 0. So τ (a0 ) = 0 and hence a0 = 0. But a0 = a ¯ generates k(y)s over k(x). So we have k(y)s = k(y) and Gal(k(x)/k(y)) = {e}. The

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homomorphism Gx → Gal(k(x)/k(y)) is trivially surjective in this case. If g ∈ Gx , then τ is the image of g under this homomorphism. So this homomorphism is always surjective. (iv) For any f ∈ B, we have Bf ∼ = (Af )G , that is, OY (D(f )) ∼ = (π∗ OX )G (D(f )). So OY → (π∗ OX )G is an isomorphism. Using this fact and (ii), one checks that Y is the quotient of X by G. Let X be a scheme on which a finite group G acts on the right. We say the action is admissible if there exists an affine morphism π : X → Y invariant under G such that OY = (π∗ OX )G . Then 3.1.1 still holds and every open subset U of Y is the quotient of π −1 (U ) by G. Proposition 3.1.2. Let X be a scheme on which a finite group G acts on the right. The following conditions are equivalent: (i) The action is admissible. (ii) X is a union of affine open subsets stable under the action of G. (iii) Each orbit of G is contained in an affine open subset. Proof. (i)⇒(iii) Let π : X → X/G be the canonical morphism. It is affine. Cover X/G by affine open subsets Ui . Then {π −1 (Ui )} is an affine open covering of X. Each orbit of G is contained in one of the affine π −1 (Ui ). (iii)⇒(ii) Let S be an orbit of G and let U be an affine open subset of X containing S. Then S ⊂ ∩g∈G gU ⊂ U . Since U is affine and S is finite, there exists an affine open subset V such that S ⊂ V ⊂ ∩g∈G gU . Indeed, suppose U = Spec A, S = {p1 , . . . , pn }, and ∩g∈G gU = Spec A − V (a), where pi are prime ideals, and a is an ideal of A. We have a 6⊂ pi . By [Matsumura (1970)] (1.B), we have a 6⊂ ∪i pi . Take f ∈ a − ∪i pi . Then V = D(f ) has the required property. Note that gV ⊂ U for all g ∈ G and gV are affine. It follows that ∩g∈G gV is affine. It is also open, stable under G, and contains S. Since S is an arbitrary orbit, X is a union of affine open subsets stable under G. (ii)⇒(i) Let Y be the quotient topological space of X by G, and let π : X → Y be the canonical continuous map. Then the ringed space (Y, (π∗ OX )G ) is a scheme and is the quotient of X by G. Indeed, if X = ∪i Ui , where each Ui = Spec Ai is affine open and stable under the action of G, then Y can be covered by the affine schemes Spec AG i . Corollary 3.1.3. Let X be a scheme on which a finite group G acts on the right.

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(i) If G acts admissibly on X, then any subgroup of G acts admissibly on X. (ii) If there exists an affine morphism f : X → Z invariant under G, then G acts admissibly on X and X/G is Z-isomorphic to Spec (f∗ OX )G . Proposition 3.1.4. Let X be a scheme on which a finite group G acts on the right and let X → Y a morphism invariant under G. For any morphism Y 0 → Y , let G act on X ×Y Y 0 by base change. (i) If G acts admissibly on X, Y ∼ = X/G, and Y 0 → Y is flat, then G 0 0 ∼ acts admissibly on X ×Y Y and Y = (X ×Y Y 0 )/G. (ii) If Y 0 → Y is quasi-compact and faithfully flat, G acts admissibly on X ×Y Y 0 , and Y 0 ∼ = (X ×Y Y 0 )/G, then G acts admissibly on X and Y ∼ = X/G. Proof. (i) We may assume X = Spec A, where A is a ring on which G acts on the left. Let B = AG . Then Y = Spec B. We may assume that Y 0 = Spec B 0 is affine, where B 0 is a flat B-algebra. The sequence Y 0→B→A→ A g∈G

is exact, where the homomorphism A →

Q

g∈G

A is defined by

a 7→ (a − ga). Since B 0 is flat over B, the sequence 0 → B 0 → A ⊗B B 0 →

Y

g∈G

(A ⊗B B 0 )

is exact. It follows that B ∼ = (A⊗B B 0 )G . So G acts admissibly on X ×Y Y 0 , 0 0 ∼ and Y = (X ×Y Y )/G. (ii) We may reduce to the case where Y = Spec B, Y 0 = Spec B 0 and B 0 is a faithfully flat B-algebra. By 1.8.7, X → Y is affine. Let X = Spec A. Then the sequence Y 0 → B 0 → A ⊗B B 0 → (A ⊗B B 0 ) 0

g∈G

0

is exact. Since B is faithfully flat over B, the sequence Y 0→B→A→ A g∈G

is exact. So B ∼ = AG . Hence G acts admissibly on X and Y ∼ = X/G.

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3.2

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([SGA 1] V 2–5.) Let X be a scheme. A geometric point of X is a morphism γ : s → X such that s is the spectrum of a separably closed field. We often write k(s) for the separably closed field such that s = Spec k(s). Giving a geometric point γ : s → X with image x ∈ X is equivalent to giving a separably closed extension k(s) of the residue field k(x). Let f : X → X 0 be a morphism. Then f γ : s → X 0 is a geometric point of X 0 . A pointed scheme is a pair (X, γ) such that X is a scheme, and γ : s → X is a geometric point of X. A morphism f : (X, γ) → (X 0 , γ 0 ) between pointed schemes is a morphism f : X → X 0 such that f γ = γ 0 . Let X be a scheme on which a finite group G acts on the right and let x ∈ X. The stabilizer Gx = {g ∈ G|gx = x} of x is called the decomposition subgroup at x and is denoted by Gd (x). This subgroup acts on the residue field k(x). The subgroup of elements in Gd (x) acting trivially on k(x) is called the inertia subgroup at x and is denoted by Gi (x). Let γ : s → X be a geometric point with image x. G acts on the set Hom(s, X). Gi (x) is exactly the stabilizer of γ ∈ Hom(s, X): Gi (x) = {g ∈ G|gγ = γ}. Using this interpretation of the inertia subgroup, one can prove the following: Proposition 3.2.1. Consider a Cartesian diagram X ×S S 0 → X ↓ ↓ S0 → S. Suppose a finite group G acts on X on the right, and suppose X → S is invariant under G. Let G act on X ×S S 0 by base change. For any x0 ∈ X ×S S 0 with image x in X, we have Gi (x) = Gi (x0 ). An etale covering space of a scheme S is a finite etale morphism X → S. The group Aut(X/S) of S-automorphisms on X acts on X on the left. Let G = Aut(X/S)◦ be the opposite group of Aut(X/S). Its elements are Sautomorphisms of X, and for any S-automorphisms g1 and g2 , the product g1 g2 in G is defined to be the composite g2 ◦ g1 . Then G acts on X on

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the right. If S = X/G, we say X → S is a galois etale covering space with galois group G. Lemma 3.2.2. Let k be a field, and let A be a finite dimensional k-algebra. Q Then A ∼ = i Li for finitely many fields Li finite separable over k if and only if the bilinear form is nondegenerate.

A × A → k,

(x, y) 7→ TrA/k (xy)

Proof. Suppose the bilinear form is nondegenerate. If x ∈ A is nilpotent, then for any y ∈ A, xy is also nilpotent, and hence Tr(xy) = 0. This implies x = 0. So A is reduced. Let K be any field containing k. Then the bilinear form (A ⊗k K) × (A ⊗k K) → K,

(x, y) 7→ Tr(xy)

is nondegenerate. So A ⊗k K is reduced. As A is a finitely dimensional kQ algebra, it is artinian and hence A = i Li for finitely many local artinian ring Li . As A is reduced, each Li must be a field. As A ⊗k K is reduced for any field K containing k, each Li is a separable extension of k. To prove the converse, it suffices to show for any field K finite separable over k, that the bilinear form K × K → k,

(x, y) 7→ Tr(xy)

is nondegenerate. Choose x ∈ K so that K is generated by x over k. Let f (t) = tn + a1 tn−1 + · · · + an be the minimal polynomial of x, and let x1 , x2 , . . . , xn be all the roots of f (t) in an algebraic closure of k. They are distinct. Since {1, x, . . . , xn−1 } is a basis of K over k, to prove that the above bilinear form is nondegenerate, it suffices to show   Tr(1 · 1) Tr(1 · x) . . . Tr(1 · xn−1 )  Tr(x · 1) Tr(x · x) . . . Tr(x · xn−1 )    det   6= 0. .. .. ..   . . . n−1 n−1 n−1 n−1 Tr(x · 1) Tr(x · x) . . . Tr(x ·x ) This determinant is equal to     1 x1 . . . xn−1 1 1 1 ... 1   1 x2 . . . xn−1 2  Y   x1 x2 . . . xn      (xi − xj )2 . det  . =  det  . . . . .    . . . . . .   .. .. . . . ..  i<j xn−1 xn−1 . . . xn−1 n 1 2 1 xn . . . xn−1 n Q Since x1 , . . . , xn are distinct, we have i<j (xi − xj )2 6= 0.

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Proposition 3.2.3. Let f : X → S be a finite flat morphism between noetherian schemes. Then A = f∗ OX is a locally free OS -module of finite rank. f is etale if and only if the bilinear form is nondegenerate.

A × A → OS ,

(x, y) 7→ Tr(xy)

Proof. We may assume that X = Spec A and S = Spec B are affine. Then as a B-module, A is finitely generated and flat. By 1.6.7, A = A∼ is a locally free OS -module of finite rank. As f is known to be flat, it is etale Q if and only if for any prime ideal q of B, Aq /qAq can be written as i Li , where Li are finitely many fields finite separable over Bq /qBq. By 3.2.2, the later condition is equivalent to saying that the bilinear form Aq /qAq × Aq /qAq → Bq /qBq,

(x, y) 7→ Tr(xy)

is nondegenerate. One then checks this last condition is equivalent to saying the bilinear form in the proposition is nondegenerate. Proposition 3.2.4. Let (S, γ) be a pointed scheme, and let (Xi , αi ) → (S, γ) (i = 1, 2) be two morphisms of pointed schemes. (i) If X1 is connected and X2 is unramified and separated over S, then there exists at most one S-morphism from (X1 , α1 ) to (X2 , α2 ). (ii) Suppose X1 and X2 are etale covering spaces of S. There exists a connected pointed etale covering space (X3 , α3 ) of (S, γ) dominating (Xi , αi ) (i = 1, 2), that is, there exist S-morphisms from (X3 , α3 ) to (Xi , αi ). Proof. (i) An S-morphism f from X1 to X2 is completely determined by its graph Γf : X1 → X1 ×S X2 , which is a section of the projection π1 : X1 ×S X2 → X1 . If f (α1 ) = α2 , then Γf (α1 ) = (α1 , α2 ). Such Γf is unique if it exists by 2.3.10 (i). (ii) α1 and α2 define a point α3 = (α1 , α2 ) of X1 ×S X2 . Let X3 be the connected component of X1 ×S X2 containing the image of α3 . Then (X3 , α3 ) dominates (Xi , αi ) (i = 1, 2). Proposition 3.2.5. Let X be a scheme on which a finite group G acts admissibly on the right and let Y = X/G be the quotient of X by G. (i) Suppose X is of finite presentation over Y . If the inertia subgroup Gi (x) is trivial for any x ∈ X, then X is etale over Y . (ii) Suppose that X is connected, S is a noetherian scheme, X → S is an etale covering space, and G ⊂ Aut(X/S)◦ . Then Gi (x) is trivial for any x ∈ X, X → Y and Y → S are etale covering spaces, and G = Aut(X/Y )◦ .

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Proof. (i) For any y ∈ Y , let OeY,¯y be the strict henselization of OY,y with respect to a separable closure of k(y). By 2.5.10, it suffices to show X ×Y Spec OeY,¯y is etale over Spec OeY,¯y . By 3.1.4, Spec OeY,¯y is the quotient of X ×Y Spec OeY,¯y by G. Making the base Spec OeY,¯y → Y , we are reduced to the case where Y = Spec A for some strictly henselian local ring A. Since X is of finite presentation and integral over Y , we have X = Spec B for some finite A-algebra B. Let n1 , . . . , nk be all the maximal ideals of B. By 2.8.3, we have B ∼ = Bn1 × · · · × Bnk . Let Gd (nj ) and Gi (nj ) be the decomposition subgroups and inertia subgroups at nj , respectively. Since A = B G , and G acts transitively on {n1 , . . . , nk } by 3.1.1 (ii), we have A∼ = (Bnj )Gd (nj ) . Since the residue field of A is separably closed, we have Gd (nj ) = Gi (nj ) = {e}. So we have A ∼ = Bnj . Thus Spec B → Spec A is etale. (ii) Let α be a geometric point of X with image x ∈ X. Any g ∈ Gi (x) induces an S-automorphism of (X, α). Since X is connected and G ⊂ Aut(X/S)◦ , we have g = e by 3.2.4 (i). So Gi (x) is trivial. Note that X → Y and Y → S are finite. To prove this, we may assume that S = Spec C is affine. Then X = Spec B for some finite C-algebra B and Y ∼ = Spec B G . Since C is noetherian, B G is finite over C. By (i), X → Y is etale. By 2.5.5, Y → S is etale. Let σ : X → X be a Y -automorphism of X. Since G acts transitively on the fibers of X → Y , there exists g1 ∈ G such that g1 (x) = σ(x). Let y be the image of x in Y . Then σ and g1 induce two k(y)-isomorphisms from k(σ(x)) to k(x), which we denote by σ \ and g1\ , respectively. We have σ \ (g1\ )−1 ∈ Gal(k(x)/k(y)). By 3.1.1 (iii), there exists g2 ∈ Gd (x) such that σ \ (g1\ )−1 is the image of g2 in Gal(k(x)/k(y)). By 3.2.4 (i), we must have σ = g2 g1 ∈ G. Hence G = Aut(X/Y )◦ . Corollary 3.2.6. Let S be a noetherian scheme, let X → S be an etale covering space, and let G be a finite group acting on the right of X such that X → S is invariant under G. Then the action is admissible and the quotient X/G is an etale covering space of S. Proof. By 3.1.3 (ii), the action is admissible. G acts on connected components of X. Let X1 , . . . , Xn be connected components of X so that any connected component of X is of the form gXj for some g ∈ G and a unique j ∈ {1, . . . , n}. Let Gj = {g ∈ G|gXj = Xj } be the stabilizer of Xj . Then ` we have X/G ∼ = j Xj /Gj . Working with each Xj and Gj , we are reduced to the case where X is connected. Replacing G by its image in Aut(X/S)◦ , we may assume G ⊂ Aut(X/S)◦ . We then apply 3.2.5 (ii).

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Proposition 3.2.7. Let (S, γ) be a pointed connected noetherian scheme, X1 and X2 two etale covering spaces of S, u : X1 → X2 an S-morphism, and Xi (γ) (i = 1, 2) the sets of geometric points of Xi lying above γ. If the map X1 (γ) → X2 (γ) induced by u is bijective, then u is an isomorphism. Proof. The image of any connected component of X2 in S is both open and closed. Since S is connected, the image is S. Replacing X2 by its connected components and X1 by the inverse images of these components, we are reduced to the case where X2 is connected. Note that u is finite and etale. So u∗ OX1 is a locally free OX2 -module of constant finite rank. Let s ∈ S be the image of γ and let x2 ∈ X2 be a point above s. Then X1 ×X2 Spec OX2 ,x2 ∼ = Spec A for some OX2 ,x2 -algebra A which is free of finite rank as an OX2 ,x2 -module. Since X1 (γ) → X2 (γ) is bijective, there is one and only one point x1 in X1 lying above x2 . So A is a local ring. Since u is etale, mx2 A is the maximal ideal of A and A/mx2 A is finite separable over OX2 ,x2 /mx2 . Again because X1 (γ) → X2 (γ) is bijective, we must have OX2 ,x2 /mx2 ∼ = A/m2 A. It follows that the rank of u∗ OX1 is 1. Let x02 be an arbitrary point of X2 and let A0 be an OX2 ,x02 -algebra such that X1 ×X2 Spec OX2 ,x02 ∼ = Spec A0 . Since rank (u∗ OX1 ) = 1, A0 is a free OX2 ,x02 -module of rank 1. The homomorphism OX2 ,x02 /mx02 → A0 /mx02 A is a nonzero homomorphism of one dimensional vectors spaces. It is necessarily surjective. By Nakayama’s lemma, the homomorphism OX2 ,x02 → A0 is also surjective. It is injective since it is faithfully flat. So we have OX2 ,x02 ∼ = u∗ OX1 , = A0 . Hence OX2 ∼ and u is an isomorphism. Proposition 3.2.8. Let (S, γ) be a pointed connected noetherian scheme, X a connected etale covering space of S, X(γ) the set of geometric points in X lying above γ, and G = Aut(X/S)◦ . The following conditions are equivalent: (i) X/G ∼ = S, that is, X is a galois covering of S. (ii) G acts transitively on X(γ). (iii) G and X(γ) have the same number of elements. Proof. Let α be a geometric point of X above γ. For any g1 , g2 ∈ G, if g1 α = g2 α, then g1 = g2 by 3.2.4 (i). It follows that #G is equal to the number of orbits of G on α. So (ii) and (iii) are equivalent.

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Let β : s → X/G be a geometric point in X/G lying above γ, let α1 , α2 : s → X be two geometric points in X lying above β, and let y, x1 , x2 be the images of β, α1 , α2 , respectively. By 3.1.1 (ii), there exists g1 ∈ G such that g1 x1 = x2 . Consider the commutative diagram k(y) → k(x1 ) ↓ &β \ ↓α\1 α\

k(x2 ) →2 k(s),

where β \ , α\1 , α\2 are the homomorphisms on residue fields induced by β, α1 , α2 , respectively. Let g1\ : k(x2 ) → k(x1 ) be the k(y)-homomorphism induced by g1 . Since k(x2 ) is a normal algebraic extension of k(y) by 3.1.1 (iii), α\1 g1\ and α\2 have the same image in k(s). So there exists τ ∈ Gal(k(x2 )/k(y)) such that α\1 g1\ τ = α\2 . By 3.1.1 (iii), there exists g2 ∈ Gx2 such that τ = g2\ . We then have α\1 (g1 g2 )\ = α\2 . Hence g1 g2 maps α1 to α2 . This shows that G acts transitively on the set X(β) of geometric points in X lying above β. So (i) implies (ii). Suppose (ii) holds. Then (X/G)(γ) has only one element since two distinct elements in (X/G)(γ) can be lifted to two elements in X(γ) that are not in the same orbit. By 3.2.5 (ii), X/G is an etale covering space of S. By 3.2.7, we have X/G ∼ = S. Proposition 3.2.9. let S be a connected noetherian scheme and let X be a connected etale covering space of S. Then any S-morphism u : X → X is an isomorphism. Proof. Note that u is necessarily etale and finite. So u(X) is both open and closed. As X is connected, we have u(X) = X. Let γ be a geometric point of S, let α be a geometric point of X lying above γ, and let x be the image of α. Then there exists x0 ∈ X such that u(x0 ) = x. Moreover the residue field k(x0 ) is a finite separable extension of the residue field k(x). So there exists a geometric point α0 of X with image x0 such that uα0 = α. Thus u induces a surjective map X(γ) → X(γ). As X(γ) is finite. This map is bijective. By 3.2.7, u is an isomorphism. Proposition 3.2.10. Let (S, γ) be a pointed connected noetherian scheme, and let (Y, β) be a pointed etale covering space of (S, γ). There exists a pointed connected galois etale covering space (X, α) of (S, γ) dominating (Y, β), that is, there exists an S-morphism from (X, α) to (Y, β). Proof. Let β1 , . . . , βk be all the distinct geometric points of Y lying above γ. They define a geometric point α of Y k = Y ×S · · · ×S Y such that

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pi α = βi (i = 1, . . . , k), where pi : Y k → Y are the projections. Let X be the connected component of Y k containing the image of α. It suffices to show that X is a galois covering of S. By 3.2.8 and 3.2.9, it suffices to show that for any geometric point α0 of X lying above γ, there exists an S-morphism u : X → X such that uα = α0 . Let j : X ,→ Y k be the open immersion. Since pi jα = βi (i = 1, . . . , k) are distinct, pi jα0 are distinct by 3.2.4 (i). Let σ be a permutation of {1, . . . , k} such that pi jα0 = pσ(i) jα, and let v : Y k → Y k be the S-morphism with the property pi v = pσ(i) . We have pi vjα = pσ(i) jα = pi jα0 . It follows that vjα = jα0 . Since X is the connected component of Y k containing the images of α and α0 , v induces a morphism u : X → X with the property u(α) = α0 . This proves our assertion. Let (S, γ) be a pointed connected noetherian scheme, and let Et(S) the category whose objects are etale covering spaces of S, and whose morphisms are S-morphisms between these covering spaces. We have a functor F from Et(S) to the category of finite sets that maps each etale covering space X of S to the set X(γ) of geometric points in X lying above γ. We call F the fiber functor for (S, γ). Let I be the opposite category of the category of pointed connected galois etale covering spaces of (S, γ). For any i ∈ ob I, denote by (Xi , αi ) the corresponding pointed connected galois etale covering space of (S, γ). Given two objects i and j in I, there exists a morphism j → i in I if there exists a morphism fij : (Xi , αi ) → (Xj , αj ). Note that such a morphism is unique if it exists by 3.2.4 (i). By 3.2.4 and 3.2.10, I satisfies the conditions (I2) and (I3) in 2.7, and for any X ∈ ob Et(S), the map lim HomS (Xi , X) → F (X), −→

i∈ob I

f 7→ f (αi ) for any f ∈ HomS (Xi , X)

is bijective. So F is pro-represented by limi∈ob I Xi . ←− Let fij : (Xi , αi ) → (Xj , αj ) be a morphism in I. We have bijections Aut(Xv /S) → F (Xv ),

σ 7→ σ(αv )

(v = i, j).

Through these bijections, the map F (Xi ) → F (Xj ),

α 7→ fij (α)

is identified with a map φij : Aut(Xi /S) → Aut(Xj /S). We have φij (σ)(αj ) = fij (σ(αi ))

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for any σ ∈ Aut(Xi /S). By 3.2.4 (i), we have φij (σ)fij = fij σ, that is, the following diagram commutes: σ

Xi → Xi fij ↓ ↓ fij φij (σ)

Xj → Xj .

One can check that φij is an epimorphism of groups. We thus get an inverse system of groups (Aut(Xi /S), φij )i∈I . We define the fundamental group of (S, γ) to be π1 (S, γ) = lim Aut(Xi /S)◦ . ←− i

◦

Note that Aut(Xi /S) acts on the left on the set Hom(Xi , X) for any object X in Et(S). So π1 (S, γ) acts on the left on the sets lim HomS (Xi , X) ∼ = F (X). −→ i

Since F (X) is finite, the map HomS (Xi , X) → F (X) is surjective for some i. Then the action of π1 (S, γ) on F (X) factors through the finite quotient Aut(Xi /S)◦ . Put the discrete topology on Aut(Xi /S)◦ , and put the product topology on π1 (S, γ) = limi Aut(Xi /S)◦ . Then π1 (S, γ) ←− acts continuously on the discrete finite set F (X). Proposition 3.2.11. An etale covering space X of a pointed connected noetherian scheme (S, γ) is connected if and only if π1 (S, γ) acts transitively on F (X). Let X1 , . . . , Xk be all the connected components of X. Then we k ` have F (X) = F (Xv ). Each F (Xv ) is stable under the action of π1 (S, γ).

Proof.

v=1

So if X is not connected, then π1 (S, γ) does not act transitively on F (X). Let α and α0 be two geometric points in X lying above γ, and let (Xi , αi ) be a pointed galois etale covering space of (S, γ) dominating (X, α). If X is connected, then the morphism (Xi , αi ) → (X, α) is surjective. So there exists a geometric point α0i of Xi lying above α0 . By 3.2.8, Aut(Xi /S) acts transitively on F (Xi ). Let σ ∈ Aut(Xi /S) such that σ(αi ) = α0i . Then σ(α) = α0 . So Aut(Xi /S) acts transitively on F (X).

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Theorem 3.2.12. Let (S, γ) be a pointed connected noetherian scheme. Then the fiber functor F : X 7→ X(γ) defines an equivalence from the category Et(S) of etale covering spaces of S to the category of finite sets on which π1 (S, γ) acts continuously on the left. Proof.

Given etale covering spaces X and Y of S, let us prove that HomS (X, Y ) → Homπ1 (S,γ) (F (X), F (Y ))

is bijective. We may assume X and Y are connected. By 3.2.4 (i), this map is injective. Fix α ∈ F (X) and β ∈ F (Y ). Let (Xi , αi ) be a pointed connected galois etale covering space of (S, γ) dominating both (X, α) and (Y, β), and let q : Xi → Y be the S-morphism with the property q(αi ) = β. Since Y is connected, q is onto. So for any φ ∈ Homπ1 (S,γ)(F (Xi ), F (Y )), there exists α0i ∈ F (Xi ) such that q(α0i ) = φ(αi ). Since Xi is galois over S, there exists σ ∈ Aut(Xi /S) such that σ(αi ) = α0i . Then qσ(αi ) = φ(αi ). Since π1 (S, γ) acts transitively on F (Xi ), qσ is mapped to φ under the map HomS (Xi , Y ) → Homπ1 (S,γ)(F (Xi ), F (Y )). Hence this map is surjective. Let Xi (α) be the set of geometric points in Xi lying above the geometric point α in X. Given two elements α1 and α2 in Xi (α), since Xi is galois over S, there exists σ ∈ Aut(Xi /S) such that σ(α1 ) = α2 . By 3.2.4 (i), σ is an X-morphism. So H = Aut(Xi /X)◦ acts transitively on Xi (α). By 3.2.8, we have X = Xi /H. For any φ ∈ Homπ1 (S,γ)(F (X), F (Y )), by the above discussion, there exists an Smorphism f 0 : Xi → Y such that f 0 (αi ) = φ(α). For any h ∈ Aut(Xi /X), we have (f 0 h)(αi ) = h(f 0 (αi )) = h(φ(α)) = φ(h(α)) = φ(α) = f 0 (αi ). By 3.2.4 (i), we must have f 0 h = f 0 . So there exists a morphism f : X → Y such that f p = f 0 , where p : Xi → X is the S-morphism such that p(αi ) = α. We have f (α) = f p(αi ) = f 0 (αi ) = φ(α). Since π1 (S, γ) acts transitively on F (X), f is mapped to φ under the map HomS (X, Y ) → Homπ1 (S,γ)(F (X), F (Y )). So this map is surjective. This proves that the functor F is fully faithful. Let A be a finite set on which π1 (S, γ) acts continuously on the left. Let us prove there exists an etale covering space X of S such that F (X) ∼ =A as sets with π1 (S, γ)-actions. Writing A as a disjoint union of orbits, and

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working with each orbit, we are reduced to the case where π1 (S, γ) acts on A transitively. Let (Xi , αi ) be a connected pointed galois covering space of S such that the action of π1 (S, γ) on A factors through Aut(Xi /S)◦ . Then there exists a subgroup H of Aut(Xi /S)◦ such that A is isomorphic to Aut(Xi /S)◦ /H. Let X = Xi /H. Then we have F (X) ∼ = A. Proposition 3.2.13. Let S be a noetherian scheme and let γ and γ 0 be two geometric points in S. If S is connected, then we have an isomorphism ∼ =

π1 (S, γ 0 ) → π1 (S, γ), and any two such isomorphisms differ by an inner automorphism of π1 (S, γ). Proof. Keep our previous notation. For each i ∈ ob I, the set of geometric points Xi (γ 0 ) in Xi lying above γ 0 is finite, nonempty, and each fij induces a surjective map from Xi (γ 0 ) to Xj (γ 0 ). We may find α0i ∈ Xi (γ 0 ) for each i such that fij (α0i ) = α0j for each fij . Let (X, α0 ) be a pointed etale covering space of (S, γ 0 ) and let X 0 be the connected component of X containing the image of α0 . Since S is connected, the morphism X 0 → S is surjective. So we can find a geometric point α of X lying above γ so that the image of α also lies in X 0 . We can find some i ∈ ob I such that (Xi , αi ) dominates (X, α). Let p : (Xi , αi ) → (X, α) be the morphism with the property p(αi ) = α. Then Xi is mapped onto X 0 and we can find a geometric point α00i of Xi over α0 . Since Xi is galois over S, there exists g ∈ Aut(Xi /S) such that g(α0i ) = α00i . Then pg is a morphism from (Xi , α0i ) to (X, α0 ). Thus any pointed etale covering space of (S, γ 0 ) is dominated by some (Xi , α0i ). This implies that the family {(Xi , α0i )} is cofinal in the category of connected pointed galois etale covering spaces of (S, γ 0 ). So we have π1 (S, γ 0 ) ∼ Aut(Xi /S)◦ ∼ = lim = π1 (S, γ). ←− i

∼ =

The isomorphism π1 (S, γ 0 ) → π1 (S, γ) depends on the choice of the family of geometric points (α0i ). Let (α00i ) be another family so that α00i are geometric points of Xi lying above γ 0 and fij (α00i ) = α00j . For each i, there exists a unique S-automorphism gi of Xi such that gi (α0i ) = α00i . One can show φij (gi ) = gj . So (gi ) is an element in π1 (S, γ). Given an element σ ∈ ∼ = π1 (S, γ 0 ), suppose its image under the isomorphism π1 (S, γ 0 ) → π1 (S, γ) defined by the family (α0i ) is (σi ), where σi ∈ Aut(Xi /S)◦ . Then the image ∼ = of σ under the isomorphism φ00 : π1 (S, γ 0 ) → π1 (S, γ) defined by the family

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(α00i ) is (gi−1 σi gi ). It follows that the two isomorphisms differ by an inner automorphism of π1 (S, γ). Proposition 3.2.14. Let K be a field, Ω a separably closed field containing K, γ : Spec Ω → Spec K the corresponding geometric point, and Ks the separable closure of K contained in Ω. Then we have a canonical isomorphism π1 (Spec K, γ) ∼ = Gal(Ks /K). Proof. Let {Ki } be the family of finite galois extensions of K contained in Ω and let αi : Spec Ω → Spec Ki be the corresponding geometric points. Then {(Spec Ki , αi )} is cofinal in the category of pointed connected galois etale covering spaces of (Spec K, γ). We have Aut(Spec Ki /Spec K)◦ = Gal(Ki /K). So we have π1 (Spec K, γ) ∼ Gal(Ki /K) ∼ = lim = Gal(Ks /K). ←− i

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([SGA 1] V 6.) Let (S 0 , γ 0 ) → (S, γ) be a morphism of pointed connected noetherian schemes, let (Xi , αi ) be the family of pointed connected galois etale covering spaces of (S, γ), and let (αi , γ 0 ) be the geometric point of Xi ×S S 0 lying above αi and γ 0 . Then (Xi ×S S 0 , (αi , γ 0 )) is a pointed etale covering space of (S 0 , γ 0 ). Let F and F 0 be the fiber functors for (S, γ) and (S 0 , γ 0 ), respectively. We have a canonical one-to-one correspondence F 0 (Xi ×S S 0 ) ∼ = F (Xi ),

through which π1 (S 0 , γ 0 ) acts on F (Xi ). We can define a map φi : π1 (S 0 , γ 0 ) → Aut(Xi /S)◦ such that φi (g 0 )(αi ) = g 0 αi for any g 0 ∈ π1 (S 0 , γ 0 ). Suppose that Xi0 is the connected component of Xi ×S S 0 containing (αi , γ 0 ). One can check that Aut(Xi0 /S) acts transitively on F 0 (Xi0 ). So Xi0 is galois over S 0 . Let gi0 be the image of g 0 ∈ π1 (S 0 , γ 0 )

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under the homomorphism π1 (S 0 , γ 0 ) → Aut(Xi0 /S 0 )◦ , and let p : Xi ×S S 0 → Xi be the projection. We have φi (g 0 )(αi ) = (p|Xi0 )(gi0 (αi , γ 0 )). On the other hand, we have φi (g 0 )(αi ) = φi (g 0 )(p|Xi0 )((αi , γ 0 )). So we have (p|Xi0 )(gi0 (αi , γ 0 )) = φi (g 0 )(p|Xi0 )((αi , γ 0 )). This implies that (p|Xi0 )gi0 = φi (g 0 )(p|Xi0 ). Using this equality, one can check that φi are homomorphisms of groups, and are compatible with the canonical homomorphisms φij : Aut(Xi /S)◦ → Aut(Xj /S)◦ whenever (Xi , αi ) dominates (Xj , αj ). So we have a continuous homomorphism φ = lim φi : π1 (S 0 , γ 0 ) → π1 (S, γ). ←− i

It follows from the definition that the following holds: Proposition 3.3.1. Let f : (S 0 , γ 0 ) → (S, γ) be a morphism of pointed connected noetherian schemes, and let φ : π1 (S 0 , γ 0 ) → π1 (S, γ) be the induced homomorphism on fundamental groups. Denote by (π1 (S 0 , γ 0 )-sets) (resp. (π1 (S, γ)-sets)) the category of finite sets on which π1 (S 0 , γ 0 ) (resp. π1 (S, γ)) acts continuously on the left, and by F 0 (resp. F ) the fiber functor for (S 0 , γ 0 ) (resp. (S, γ)). Consider the functor Hf : Et(S) → Et(S 0 ),

X 7→ X ×S S 0

and the functor Hφ : (π1 (S, γ)-sets) → (π1 (S 0 , γ 0 )-sets) induced by the homomorphism φ. The following diagram commutes: Et(S) F ↓

Hf

→

Hφ

Et(S 0 ) ↓ F0

(π1 (S, γ)-sets) → (π1 (S 0 , γ 0 )-sets).

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Suppose that (S 0 , γ 0 ) → (S, γ) is an etale covering space. Let {(Xi , αi )} be the family of pointed connected galois etale covering spaces of (S, γ) dominating (S 0 , γ 0 ). By 3.2.4 and 3.2.10, {(Xi , αi )} is cofinal in the category of pointed connected galois etale covering spaces of (S, γ). So we have π1 (S, γ) ∼ Aut(Xi /S)◦ . = lim ←− i

0 ◦

One can show that Aut(Xi /S ) acts transitively on the set Xi (γ 0 ) of geometric points in Xi lying above γ 0 . So Xi are galois over S 0 . Again by 3.2.4 and 3.2.10, {(Xi , αi )} is cofinal in the category of pointed connected galois etale covering spaces of (S 0 , γ 0 ). So we have π1 (S 0 , γ 0 ) ∼ Aut(Xi /S 0 )◦ . = lim ←− i

0

0

The homomorphism π1 (S , γ ) → π1 (S, γ) can be identified with the canonical inclusion lim Aut(Xi /S 0 )◦ → lim Aut(Xi /S)◦ . ←− ←− i

0

i

0

In particular, π1 (S , γ ) → π1 (S, γ) is injective. A topological group π is called a profinite group if it is isomorphic to a topological group of the form limi∈I Gi , where I is a direct set, and {Gi } ←− is an inverse system of finite groups with discrete topology, and limi∈I Gi ←− is provided with the product topology. Confer 4.2 for more properties of profinite groups. Lemma 3.3.2. Let φ : π 0 → π be a continuous homomorphism of profinite groups, (π 0 -sets) (resp. (π-sets)) the category of finite sets on which π (resp. π 0 ) acts continuously on the left, and Hφ : (π-sets) → (π 0 -sets) the functor induced by φ. (i) φ is surjective if and only if for any A ∈ ob (π-sets) on which π acts transitively, π 0 acts transitively on Hφ (A). (ii) φ is trivial if and only if for any A ∈ ob (π-sets), π acts trivially on Hφ (A). (iii) Let H be an open subgroup of π. Then im φ ⊂ H if and only if the element eH in Hφ (π/H) is fixed by π 0 . (iv) Let H 0 be an open subgroup of π 0 . Then ker φ ⊂ H 0 if and only if there exists an object A in (π-sets) on which π acts transitively such that there exists a π 0 -morphism from the π 0 -orbit of an element in Hφ (A) to π 0 /H 0 . If ker φ ⊂ H 0 and φ is surjective, then there exists an object A in (π-sets) on which π acts transitively and π 0 /H 0 ∼ = Hφ (A).

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(v) φ is injective if and only if for any A0 ∈ ob (π 0 -sets), there exists A ∈ ob (π-sets) on which π acts transitively such that there exists a π 0 morphism from the π 0 -orbit of an element in Hφ (A) to A0 . Proof. (i) Let us prove the “if” part. Write π = limi πi , where πi are finite ←− groups such that π → πi are surjective. Then π acts transitively on πi . If π 0 acts transitively on Hφ (πi ), then the composites φ

π 0 → π → πi are surjective. Hence im φ is dense in π. But φ is a continuous map from a compact space to a Hausdorff space. So φ is a closed mapping. It follows that im φ = π. (ii) and (iii) are obvious. T (iv) Suppose ker φ ⊂ H 0 . We have {e} = H, where H goes over the TH set of open subgroups of π. It follows that φ−1 (H) = ker φ ⊂ H 0 , that H S is, H 0c ⊂ φ−1 (H)c . Since H 0c is compact and φ−1 (H)c are open, there H

exist finitely many open subgroups H1 , . . . , Hk of π such that H 0c ⊂ φ−1 (H1 )c ∪ · · · ∪ φ−1 (Hk )c .

Let H = H1 ∩ · · · ∩ Hk . Then we have φ−1 (H) ⊂ H 0 . Let A = π/H and let A0 be the π 0 -orbit of eH in Hφ (A). Then A0 → π 0 /H 0 ,

φ(g 0 )H 7→ g 0 H 0 for any g 0 ∈ π 0

is a well-defined π 0 -morphism. Suppose that ker φ ⊂ H 0 and φ is surjective. Then we have π 0 /H 0 ∼ = π/φ(H 0 ). In particular, φ(H 0 ) has finite index in π. Since φ(H 0 ) is compact, and hence closed in π, it is open. Take A = π/φ(H 0 ). Then we have π 0 /H 0 ∼ = Hφ (A). Suppose that A is an object in ob (π-sets) on which π acts transitively such that there exists a π 0 -morphism from the π 0 -orbit of an element a in A to π 0 /H 0 . We may choose a so that it is mapped to eH 0 in π 0 /H 0 . Let H ⊂ π be the stabilizer of a. Then H is an open subgroup of π, and the π 0 orbit of a is isomorphic to π 0 /φ−1 (H) so that a is identified with eφ−1 (H). So we have a π 0 -morphism π 0 /φ−1 (H) → π 0 /H 0 mapping eφ−1 (H) to eH 0 . This is possible only if φ−1 (H) ⊂ H 0 . In particular, we have ker φ ⊂ H 0 . (v) follows from (iv).

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Lemma 3.3.3. Let φ : π 0 → π and ψ : π → π 00 be two continuous homomorphisms of profinite groups, (π-sets) (resp. (π 0 -sets), resp. (π 00 -sets)) the category of finite sets on which π (resp. π 0 , resp. π 00 ) acts continuously on the left, and Hφ : (π-sets) → (π 0 -sets) (resp. Hψ : (π 00 -sets) → (π-sets)) the functor induced by φ (resp. ψ). (i) ψφ is trivial if and only if for any object A00 in (π 00 -sets), π 0 acts trivially on Hφ Hψ (A00 ). (ii) kerψ ⊂ imφ if and only if for any object A in (π-sets) such that π acts transitively and that there exists an element in A fixed by im φ, there exists an object A00 in (π 00 -sets) on which π 00 acts transitively such that there exists a π-morphism from the π-orbit of an element in Hψ (A00 ) to A. Proof. (i) follows from 3.3.2 (ii). (ii) follows from 3.3.2 (iii) and (iv) and the fact that ker ψ ⊂ im φ if and only if any open subgroup H of π containing im φ contains ker ψ. Indeed, for any closed subgroup C of a T profinite group G, we have C = H, where H goes over the family of H⊃C

open subgroups of G containing C. This can be proved as follows. If we have x 6∈ C, then by 4.2.6 in the next chapter, there exists a normal open subgroup N of G such that xN ∩ C = Ø. Then H = N C is an open subgroup of G containing C and x 6∈ H. ¿From 3.3.1–3.3.3, we get the following two propositions:

Proposition 3.3.4. Let (S 0 , γ 0 ) → (S, γ) be a morphism of pointed connected noetherian schemes. (i) π1 (S 0 , γ 0 ) → π1 (S, γ) is surjective if and only if for any connected etale covering space X of S, X ×S S 0 is connected. (ii) π1 (S 0 , γ 0 ) → π1 (S, γ) is injective if and only if for any connected etale covering space X 0 of S 0 , there exists a connected etale covering space X of S such that there exists an S 0 -morphism from a connected component of X ×S S 0 to X 0 . (iii) π1 (S 0 , γ 0 ) → π1 (S, γ) is trivial if and only if for any connected etale covering space X of S, X ×S S 0 is S 0 -isomorphic to a disjoint union of copies of S 0 . Proposition 3.3.5. Let (S 0 , γ 0 ) → (S, γ) → (S 00 , γ 00 ) be morphisms of pointed connected noetherian schemes, and let φ

ψ

π1 (S 0 , γ 0 ) → π1 (S, γ) → π1 (S 00 , γ 00 )

be the corresponding homomorphisms. (i) ψφ is trivial if and only if for any connected etale covering space X 00 of S 00 , X 00 ×S 00 S 0 is S 0 -isomorphic to a disjoint union of copies of S 0 .

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(ii) kerψ ⊂ imφ if and only if for any connected etale covering space X of S such that X ×S S 0 → S 0 admits a section, there exists a connected etale covering space X 00 of S 00 and an S-morphism from a connected component of X 00 ×S 00 S to X. Proposition 3.3.6. Let S be a normal connected noetherian scheme, K its function field, Ω a separably closed field containing K, and Ks the separable closure of K in Ω. Denote by γ both the geometric point Spec Ω → Spec K and the geometric point defined by the composite Spec Ω → Spec K → S. Then the canonical homomorphism π1 (Spec K, γ) → π1 (S, γ)

is an epimorphism. Through the isomorphism π1 (Spec K, γ) ∼ = Gal(Ks /K), the kernel of the above epimorphism is identified with the subgroup Gal(Ks /Kur ), where Kur is the subfield of Ks generated by all finite separable extensions of K contained in Ks that are unramified over S. In particular, we have π1 (S, γ) ∼ = Gal(Kur /K). Proof. Let X be a connected etale covering space of S. Then X is normal and integral. So X ⊗S K is integral and hence connected. By 3.3.4 (i), the homomorphism π1 (Spec K, γ) → π1 (S, γ) is surjective. Let {(Xi , αi )} be the family of pointed connected galois etale covering spaces of (S, γ), and let Ki be the function field of Xi . Then Ki are finite galois extensions of K unramified over S, and Aut(Xi /S)◦ ∼ = Gal(Ki /K) by 3.1.4 (i). The geometric points αi define embeddings of Ki into Ω. By 2.9.2, Kur is generated by the images of Ki in Ω, and we have π1 (S, γ) ∼ Gal(Ki /K) ∼ = lim = Gal(Kur /K). ←− i

Proposition 3.3.7. Let k be a field, ks a separable closure of k, S a connected k-scheme of finite type, α a geometric point of S ⊗k ks , and β (resp. γ) the geometric points in S (resp. Spec k) defined by the composite of α with S ⊗k ks → S (resp. S ⊗k ks → Spec k). Suppose S ⊗k ks is connected. Then the following sequence is exact: 1 → π1 (S ⊗k ks , α) → π1 (S, β) → π1 (Spec k, γ) → 1. Proof. Any connected etale covering space of Spec k is isomorphic to Spec K → Spec k for some finite separable extension K of k contained in

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ks . Since S ⊗k ks is connected and the projection S ⊗k ks → S ⊗k K is surjective, S ⊗k K is also connected. By 3.3.4 (i), π1 (S, β) → π1 (Spec k, γ) is surjective. We have (S ⊗k ks ) ⊗k K ∼ = S ⊗k (ks ⊗k K). Since ks ⊗k K is a direct product of finitely many copies of ks , (S ⊗k ks )⊗k K is a disjoint union of copies of S ⊗k ks . By 3.3.5 (i), the composite π1 (S ⊗k ks , α) → π1 (S, β) → π1 (Spec k, γ) is trivial. Let X → S be a connected etale covering space such that X ⊗k ks → S ⊗k ks admits a section. Then by 1.10.9, there exists a finite separable extension K of k contained in ks such that X ⊗k K → S ⊗k K admits a section. Taking the composite of this section with the projection X ⊗k K → X, we get an S-morphism S ⊗k K → X. By 3.3.5, the sequence π1 (S ⊗k ks , α) → π1 (S, β) → π1 (Spec k, γ) is exact. Let X 0 → S⊗k ks be a connected etale covering space. By 1.10.9, 1.10.10 and 2.3.7, there exist a finite separable extension K of k contained in ks and an etale covering space X1 → S ⊗k K inducing X 0 → S ⊗k ks by the base extension K → ks . Since the projection X 0 → X1 is surjective, X1 is connected. The composite X1 → S ⊗ k K → S is an etale covering space of S. The graph Γ : X1 → X1 ×S (S ⊗k K) ∼ = X1 ⊗ k K of the S-morphism X1 → S ⊗k K is a section of the projection X1 ⊗k K → X1 . It induces an isomorphism of X1 with a connected component of X1 ⊗k K. The inverse of this isomorphism is an (S ⊗k K)-isomorphism from a connected component of X1 ⊗k K to X1 . By the base extension K → ks , we get an (S ⊗k ks )-morphism from a connected component of X1 ⊗k ks to X 0 . By 3.3.4 (ii), π1 (S ⊗k ks , α) → π1 (S, β) is injective. Γ

X1 → X1 ⊗ k K →

X1 ↓ & ↓ S ⊗k K ↓ S ⊗k K → S

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Chapter 4

Group Cohomology and Galois Cohomology

4.1

Group Cohomology

([Serre (1979)] VII, VIII.) Let G be a group. An abelian group with a left G-action can be identified with a Z[G]-module. We also call a Z[G]-module a G-module. A homomorphism of G-modules is defined to be a homomorphism of Z[G]-modules. It is a homomorphism of abelian groups compatible with the G-actions. For any G-module A, define AG = {a ∈ A|ga = a for all g ∈ G},

AG = A/the subgroup generated by {ga − a|g ∈ G, a ∈ A}.

We call AG (resp. AG ) the group of G-invariants (resp. G-coinvariants) of A. It is the maximal subgroup (resp. quotient group) of A on which G acts trivially. The functor A 7→ AG on the category of G-modules is denoted by ΓG . It is left exact, and its i-th derived functor is denoted by H i (G, −) or Ri ΓG . Obviously we have AG = HomZ[G] (Z, A), where Z is the group of integers with the trivial G-action. We thus have H i (G, A) ∼ = ExtiZ[G] (Z, A).

So if I · (resp. P· ) is an injective (resp. projective) resolution of A (resp. Z) in the category of G-modules, then we have ∼ H i (I ·G ) ∼ H i (G, A) = = H i (HomG (P· , A)). Proposition 4.1.1. Let G = Z. For any  G A H i (G, A) ∼ = AG  0 139

G-module A, we have if i = 0, if i = 1, if i ≥ 2.

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Proof. Let T correspond to the canonical generator 1 of the group G = Z. We can identify Z[G] with Z[T, T −1 ]. Consider the exact sequence T −1

0 → Z[T, T −1 ] → Z[T, T −1 ] → Z → 0,

where T − 1 : Z[T, T −1 ] → Z[T, T −1 ] is the multiplication by T − 1, and P P ( i ai T i ) = i ai . This gives rise to a resolution of Z by free Z[G]modules. So H i (G, A) is the i-th cohomology group of the complex T −1

0 → HomZ[G] (Z[G], A) → HomZ[G] (Z[G], A) → 0. This complex can be identified with T −1

0 → A → A → 0, where T − 1 : A → A is the homomorphism defined by (T − 1)(a) = T a − a. Our assertion follows. Proposition 4.1.2. Let G = Z/n for some n ∈ N. For any G-module A, let T : A → A be the homomorphism defined by the action of the canonical generator ¯ 1 ∈ Z/n. We have  G if i = 0, A H i (G, A) ∼ = ker(T n−1 + · · · + T + 1)/im(T − 1) if i = 1, 3, 5, . . . ,  ker(T − 1)/im(T n−1 + · · · + T + 1) if i = 2, 4, 6, . . . .

Proof. We can identify Z[G] with Z[T ]/(T n − 1). We have an exact sequence T −1

· · · → Z[G]

T n−1 +···+T +1

→

T −1

Z[G] → Z[G]

T n−1 +···+T +1

→

T −1

Z[G] → Z[G] → Z → 0,

where T − 1 : Z[G] → Z[G] (resp. T n−1 + · · · + T + 1 : Z[G] → Z[G]) is the P P multiplication by T − 1 (resp. T n−1 + · · · + T + 1), and ( g ag g) = g ag . This gives rise to a resolution of Z by free Z[G]-modules. We use this resolution to calculate H i (G, A) ∼ = ExtiZ[G] (Z, A). Let Li be the free abelian group generated by elements (g0 , . . . , gi ) in G (i ≥ 0). Define a left G-action on Li by i+1

g(g0 , . . . , gi ) = (gg0 , . . . , ggi ). Then Li is a free Z[G]-module with a basis consisting of elements of the form (1, g1 , g1 g2 , . . . , g1 g2 · · · gi ). Define a G-homomorphism ∂i : Li → Li−1 for each i ≥ 1 by ∂i (g0 , . . . , gi ) =

i X j=0

(−1)j (g0 , . . . , gˆj , . . . , gi ),

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and a G-homomorphism : L0 → Z by (g0 ) = 1. Then the sequence ∂

∂

· · · → Li →i Li−1 → · · · → L1 →1 L0 → Z → 0 is exact. Indeed, fix an element g ∈ G, then the homomorphisms hi : Li → Li+1 ,

h−1 : Z → L0 ,

hi (g0 , . . . , gi ) = (g, g0 , . . . , gi ), h−1 (1) = (g)

define a homotopy between the identity morphism and the zero morphism of the above complex. Thus L· is a resolution of Z by free Z[G]-modules, and for any Z[G]-module A, H i (G, A) is the i-th cohomology group of the complex d

d

0 → HomZ[G] (L0 , A) →0 HomZ[G] (L1 , A) →1 · · · d

→ HomZ[G] (Li , A) →i HomZ[G] (Li+1 , A) → · · · . Let C i (G, A) be the set of all maps from Gi to A. We have isomorphisms Fi : HomZ[G] (Li , A) → C i (G, A) defined by Fi (φ)(g1 , . . . , gi ) = φ(1, g1 , g1 g2 , . . . , g1 g2 · · · gi ) for any φ ∈ HomZ[G] (Li , A) and (g1 , . . . , gi ) ∈ Gi . Through these isomorphisms, the homomorphisms di : HomZ[G] (Li , A) → HomZ[G] (Li+1 , A) are identified with the homomorphisms di : C i (G, A) → C i+1 (G, A) defined by (di f )(g1 , . . . , gi+1 ) = g1 f (g2 , . . . , gi+1 ) +

i X

(−1)j f (g1 , . . . , gj−1 , gj gj+1 , gj+2 , . . . , gi+1 )

j=1

i+1

+(−1)

f (g1 , . . . , gi )

for any f ∈ C i (G, A) and (g1 , . . . , gi+1 ) ∈ Gi+1 . So H i (G, A) is the i-th cohomology group of the complex d

d

0 → C 0 (G, A) →0 C 1 (G, A) → · · · → C i (G, A) →i C i+1 (G, A) → · · · .

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A 1-cocycle of the complex C · (G, A) is a map f : G → A satisfying the condition f (g1 g2 ) = g1 f (g2 ) + f (g1 ) for any g1 , g2 ∈ G. A 1-coboundary of the complex C · (G, A) is a map f : G → A such that there exists a ∈ A satisfying the condition f (g) = ga − a

for any g ∈ G. If G acts trivially on A, we have H 1 (G, A) = Hom(G, A).

Let f : G0 → G be a homomorphism of groups, A a G-module, A0 a G -module, and φ : A → A0 an additive map compatible with the group actions, that is, we have 0

φ(f (g 0 )a) = g 0 φ(a) 0

for any g 0 ∈ G0 and a ∈ A. Then φ induces a homomorphism AG → A0G . By the universal property of derived functors, it extends canonically to a family of homomorphisms H i (G, A) → H i (G0 , A0 ). In particular, if H is a subgroup of G and A is a G-module, we can take f : G0 → G to be the inclusion H ,→ G, and φ : A → A0 to be id : A → A. We thus get the restriction homomorphisms Res : H i (G, A) → H i (H, A).

If H is normal in G, we can take f : G0 → G to be the projection G → G/H, and φ : A → A0 to be the inclusion AH ,→ A. We thus get the inflation homomorphisms Inf : H i (G/H, AH ) → H i (G, A). Let H be a subgroup of G and let B be an H-module. Define IndG H B = HomZ[H] (Z[G], B). Note that IndG H B can be identified with the set of maps φ : G → B satisfying φ(hg) = hφ(g) for any h ∈ H and g ∈ G. Define a G-action on IndG H B by (gφ)(g 0 ) = φ(g 0 g)

0 for any φ ∈ IndG H B and g, g ∈ G. Let

θ : IndG HB → B

be the homomorphism defined by θ(φ) = φ(1) for any φ ∈ IndG H B. It is compatible with the group actions.

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Theorem 4.1.3 (Shapiro). For any H-module B, θ induces isomor∼ = i phisms H i (G, IndG H B) → H (H, B) for all i. Proof.

For any G-module A, one can check that the map HomG (A, IndG H B) → HomH (A, B),

f 7→ θ ◦ f

is bijective. In other words, the functor B 7→ IndG H B from the category of H-modules to the category of G-modules is right adjoint to the canonical forgetful functor A 7→ A from the category of G-modules to the category of H-modules. As the forgetful functor is exact, for any injective H-module G B, IndG H B is an injective G-module. But the functor B 7→ IndH B is exact G G ∼ H and θ induces an isomorphism (IndH B) = B . It follows that θ induces ∼ i isomorphisms H i (G, IndG H B) = H (H, B) for all i. Consider the case where H = {1}. Any abelian group M can be considered as an H-module. We can form the G-module IndG {1} M = Hom(Z[G], M ). By 4.1.3, we have i H i (G, IndG {1} M ) = H ({1}, M ).

But H i ({1}, M ) = 0 for any i ≥ 1. So we have H i (G, IndG {1} M ) = 0 for any i ≥ 1. A G-module is called induced if it is isomorphic to IndG {1} M for some abelian group M . A G-module is called weakly injective if it is a direct factor of an induced G-module. For any weakly injective G-module I, we have H i (G, I) = 0 for any i ≥ 1. Any G-module A can be embedded into an induced G-module. Indeed, the homomorphism ι : A → IndG {1} A = Hom(Z[G], A),

ι(a)(g) = ga

is a G-monomorphism. Injective G-modules are weakly injective. We can use resolutions of A by weakly injective G-modules to calculate H i (G, A). Let H be a subgroup of G. Then any induced G-module is also an induced H-module. Indeed, for any abelian group M , we have M Z[gH], M ) IndG {1} M = Hom(Z[G], M ) = Hom( ∼ =

Y

gH∈G/H

gH∈G/H

Hom(Z[gH], M ) ∼ = Hom(Z[H],

Y

gH∈G/H

M ).

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So any weakly injective G-module is also a weakly injective H-module. In particular, any injective G-module I is a weakly injective H-module, and we have H i (H, I) = 0 for any i ≥ 1. If 0 → I0 → I1 → · · ·

is a resolution of a G-module A by injective G-modules, then H i (H, A) = H i (I ·H ). Suppose H is normal in G. Then for any G-module A, AH is a G/H-module and we have (AH )G/H = AG . If I is an injective G-module, then I H is an injective G/H-module. Using these facts, one can construct a biregular spectral sequence E2ij = H i (G/H, H j (H, A)) ⇒ H i+j (G, A), which is called the Hochschild–Serre spectral sequence. Note that for any g ∈ G, the action of gH ∈ G/H on H j (H, A) is induced by the group isomorphism H → H,

h 7→ g −1 hg

and the additive map A → A,

a 7→ ga

which is compatible with the group actions. Again let H be a subgroup of G. For any G-module A, the Ghomomorphism ι : A → IndG H A = HomZ[H] (Z[G], A),

ι(a)(g) = ga

induces homomorphisms H i (G, A) → H i (G, IndG H A).

∼ i Composed with the isomorphisms H i (G, IndG H A) = H (H, A), the resulting homomorphisms H i (G, A) → H i (H, A) coincide with the restriction homomorphisms. Suppose H has finite index in G. We have a G-homomorphism X f 7→ gf (g −1 ). π : IndG H A = HomZ[H] (Z[G], A) → A, gH∈G/H

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(Note that gf (g −1 ) depends on the left coset gH, not on the choice of g.) It induces homomorphisms i H i (G, IndG H A) → H (G, A).

∼ i Composed with the isomorphisms H i (G, IndG H A) = H (H, A), the resulting homomorphisms Cor : H i (H, A) → H i (G, A) are called the corestriction homomorphisms. Proposition 4.1.4. Suppose H is a subgroup of G with finite index n. Then we have Cor ◦ Res = n. Proof.

Notation as above. The composite ι

π

A → IndG HA → A coincides with multiplication by n. Our assertion follows.

Corollary 4.1.5. Let G be a finite group, p a prime number, Gp a Sylow p-subgroup of G, and A a G-module. Then Res : H i (G, A) → H i (Gp , A) is injective on the p-primary part of H i (G, A) for any i. In particular, if H i (Gp , A) = 0 for each prime number p, then H i (G, A) = 0. Proof. have

Let x ∈ H i (G, A) be an element in the kernel of Res. Then we [G : Gp ]x = Cor ◦ Res(x) = 0.

If x lies in the p-primary part of H i (G, A), then pk x = 0 for some natural number k. As [G : Gp ] is relatively prime to p, this implies that x = 0. Corollary 4.1.6. Let G be a finite group of order n. Then for any Gmodule A and any i ≥ 1, H i (G, A) is annihilated by n. Proof.

Apply 4.1.4 to the subgroup H = {1}.

Corollary 4.1.7. Let G be a finite group and let A be a G-module finitely generated as an abelian group. Then H i (G, A) are finite for all i ≥ 1. Proof. Since H i (G, A) are the cohomology groups of the complex C · (G, A), they are finitely generated abelian groups. By 4.1.6, H i (G, A) are torsion groups for all i ≥ 1. So they are finite.

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Etale Cohomology Theory

Profinite Groups

([Serre (1964)] I 1.) Lemma 4.2.1. Let X be a compact Hausdorff topological space and let x ∈ X. The connected component of X containing x is ∩λ Uλ , where {Uλ } is the family of compact open neighborhoods of x. Proof. Let C be the connected component of X containing x. Any compact open neighborhood Uλ of x is both open and closed, and hence contains C. So we have C ⊂ ∩λ Uλ . To prove our assertion, it suffices to show that ∩λ Uλ is connected. Let A ⊂ ∩λ Uλ be open and closed in ∩λ Uλ . Then A and ∩λ Uλ − A are closed in X. We can find disjoint open subsets U and V in X such that A ⊂ U,

∩λ Uλ − A ⊂ V.

We have ∩λ Uλ ⊂ U ∪ V and hence X − U ∪ V ⊂ ∪λ (X − Uλ ). But X − U ∪ V is compact. So X − U ∪ V ⊂ (X − Uλ1 ) ∪ · · · ∪ (X − Uλn ) for finitely many compact open neighborhoods Uλi (i = 1, . . . , n) of x. Then we have Uλ1 ∩ · · · ∩ Uλn ⊂ U ∪ V. So Uλ1 ∩ · · · ∩ Uλn is the disjoint union of the open subsets Uλ1 ∩ · · · ∩ Uλn ∩ U and Uλ1 ∩ · · · ∩ Uλn ∩ V . Hence Uλ1 ∩ · · · ∩ Uλn ∩ U and Uλ1 ∩ · · · ∩ Uλn ∩ V are open and closed. As X is compact, Uλ1 ∩ · · ·∩ Uλn ∩ U and Uλ1 ∩ · · · ∩ Uλn ∩ V are also compact. Only one of them contains x. If x ∈ Uλ1 ∩· · ·∩Uλn ∩U , then Uλ1 ∩· · ·∩Uλn ∩U is a compact open neighborhood of x, and hence ∩λ Uλ ⊂ Uλ1 ∩ · · · ∩ Uλn ∩ U . This implies that ∩λ Uλ − A ⊂ U . But ∩λ Uλ − A ⊂ V , and U and V are disjoint. So we must have ∩λ Uλ − A = Ø. If x ∈ Uλ1 ∩ · · · ∩ Uλn ∩ V , then ∩λ Uλ ⊂ Uλ1 ∩ · · · ∩ Uλn ∩ V . This implies that A ⊂ V . But A ⊂ U , and U and V are disjoint. So we must have A = Ø. This shows that ∩λ Uλ is connected. Lemma 4.2.2. Let G be a compact topological group. Then G is totally disconnected if and only if ∩λ Uλ = {1}, where {Uλ } is the family of compact open neighborhoods of 1. Recall that a topological space X is called totally disconnected if any connected subset of X contains only one point. Proof. Suppose that G is compact and totally disconnected. Since {1} is connected, we have {1} = {1}. Hence {1} is a closed subset of G. Since

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the diagonal ∆ = {(x, x)|x ∈ G} is the inverse image of {1} under the continuous map G × G → G,

(x, y) 7→ xy −1 ,

it is closed and hence G is Hausdorff. By 4.2.1, we must have ∩λ Uλ = {1}. Conversely, suppose ∩λ Uλ = {1}. For any g ∈ {1}, as gUλ−1 is an open neighborhood of g, we have 1 ∈ gUλ−1 , and hence g ∈ Uλ . So we have g ∈ ∩λ Uλ = {1}. This shows that {1} is closed. As above, this implies that G is Hausdorff. By 4.2.1, G is totally disconnected. Lemma 4.2.3. Let G be a Hausdorff topological group and let U be a compact open neighborhood of {1}. Then U contains a compact open subgroup of G. Proof. Let F = (G − U ) ∩ U 2 . Then F is closed. We have U · 1 ⊂ G − F . Since U is compact, we can find an open neighborhood V of 1 contained in U such that U V ⊂ G − F . Replacing V by V ∩ V −1 , we may assume V = V −1 . We have U V ⊂ (G − F ) ∩ U 2 ⊂ U . From this, we get U V n ⊂ U ∞ S for all n ≥ 1. Let H = V n . Then H is an open subgroup of G. Any n=1

open subgroup is necessarily closed. As H is contained in U , H must be compact. Lemma 4.2.4. Let G be a totally disconnected group and let U be a compact neighborhood (not necessarily open) of 1. Then U contains a compact open subgroup of G.

Proof. Let V be an open neighborhood of 1 contained in U . Since G is totally disconnected, it is Hausdorff by the proof of 4.2.2. By 4.2.1, we have {1} = ∩λ Uλ , where {Uλ } is the family of compact open neighborhoods of 1 in U . We have ∩λ Uλ = {1} ⊂ V . Hence U − V ⊂ ∪λ (U − Uλ ). But U − V is compact. So U − V ⊂ (U − Uλ1 ) ∪ · · · ∪ (U − Uλn ) for finitely many compact open neighborhoods Uλi (i = 1, . . . , n) of 1 in U . Then we have Uλ1 ∩ · · · ∩ Uλn ⊂ V. Note that Uλ1 ∩ · · · ∩ Uλn is open in V and hence in G, and it is compact. By 4.2.3, Uλ1 ∩ · · · ∩ Uλn contains a compact open subgroup of G. Lemma 4.2.5. Let G be a topological group, K a compact subset of G, and U a neighborhood of 1. Then there exists a neighborhood V of 1 such that xV x−1 ⊂ U for any x ∈ K.

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The continuous map G × G → G,

(x, y) 7→ xyx−1

maps (x, 1) to 1. So for any x ∈ K, there exists a neighborhood Wx of x and a neighborhood Wx0 of 1 such that x0 yx0−1 ∈ U for any x0 ∈ Wx and y ∈ Wx0 . Let {Wx1 , . . . , Wxn } be a finite covering of K. We can take V = Wx0 1 ∩ · · · ∩ Wx0 n . Proposition 4.2.6. Let G be a compact totally disconnected group and let U be a neighborhood of 1. Then U contains a normal open subgroup of G. Proof. Since G is compact and Hausdorff, U contains a compact neighborhood (not necessarily open) of 1. By 4.2.4, U contains an open subgroup of V . By 4.2.5, there exists a neighborhood V 0 of 1 such that xV 0 x−1 ⊂ V T −1 T −1 for any x ∈ G. Then V 0 ⊂ x V x. Note that x V x is an open x∈G

normal subgroup of G contained in U .

x∈G

Recall that a topological group is called profinite if it is isomorphic to the inverse limit of an inverse system of finite discrete groups. Corollary 4.2.7. (i) A topological group is profinite if and only if it is compact and totally disconnected. (ii) A closed subgroup of a profinite group is profinite. (iii) Direct products of profinite groups are profinite. (iv) The inverse limit of any inverse system of profinite groups is profinite. (v) If H is a closed subgroup of a profinite group, then the space of left cosets G/H with the quotient topology is totally disconnected. If H is a normal closed subgroup of G, then G/H is profinite. Proof. (i) Let G = limi Gi be a profinite group, where {Gi } is an inverse system ←− Q of finite discrete groups. Note that limi Gi is a closed subset of i Gi . By ←− Tychonoff’s Theorem, G is compact. Let C be a connected subset of G. Then for each i, the image of C under the projection G → Gi is connected, and hence has only one element since Gi is discrete. It follows that C has only one element. So G is totally disconnected. Conversely, let G be a compact totally disconnected group. By 4.2.6, open normal subgroups of G form a base of neighborhoods of 1. For any open normal subgroup N of G, the group G/N is a compact discrete group

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and hence finite. Consider the profinite group limN G/N , where N goes over ←− the family of open normal subgroups of G. The canonical homomorphism G → limN G/N is continuous and has dense image. As G is compact and ←− limN G/N is Hausdorff, this homomorphism is a closed map. By 4.2.2, this ←− homomorphism is injective. These facts imply that G → limN G/N is an ←− isomorphism of topological groups. So G is profinite. (ii) and (iii) follow from (i). (iv) follows from (ii) and (iii) since the inverse limit of an inverse system of topological groups is isomorphic to a closed subgroup of the product of these topological groups. (v) By 4.2.1, to show that G/H is totally disconnected, it suffices to show that for any g ∈ G, we have {gH} = ∩λ Uλ , where {Uλ } is the family of compact open neighborhoods of gH in G/H. If xH 6∈ {gH}, that is, if 1 6∈ g −1 xH, then there exists an open subgroup U disjoint from g −1 xH. The canonical homomorphism π : G → G/H is continuous and open. Since U is an open subgroup, it is necessarily closed and hence compact. It follows that π(gU ) is a compact open neighborhood of gH in G/H. We have xH 6∈ π(gU ). So xH 6∈ ∩λ Uλ . Our assertion follows. Q np A surnatural number is a formal product p p , where p goes over the set of prime numbers, and np ∈ N ∪ {0, ∞}. One defines the product, the greatest common divisor, and the least common multiple of a family of surnatural numbers in the obvious way. Let G be a profinite group and let H be a closed subgroup of G. The index [G : H] of H in G is defined to be the least common multiple of [G/U : H/H ∩ U ], where U goes over the family of open normal subgroups of G. It is a surnatural number. It is also the least common multiple of [G : V ], where V goes over the family of open subgroups of G containing H. Indeed, if U is an open normal subgroup, then HU is an open subgroup containing H, and [G/U : H/H ∩ U ] = [G : HU ]; if V is an open subgroup containing H = H · 1, then HU ⊂ V for some open normal subgroup U and [G : V ] [G : HU ] = [G/U : H/H ∩ U ].

Proposition 4.2.8. Let G be a profinite group. (i) For all closed subgroups K and H of G such that K ⊂ H, we have [G : K] = [G : H][H : K]. (ii) For all closed subgroups K and H of G such that K ⊂ H and K is normal in G, we have [G/K : H/K] = [G : H].

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(iii) A closed subgroup H in G is open if and only if [G : H] is finite. (iv) If {Hi } is a decreasing filtered family of closed subgroups in G and H = ∩i Hi , then [G : H] is the least common multiple of [G : Hi ]. Proof. (i) For any open normal subgroup U of G, we have

[G/U : K/K ∩U ] = [G/U : H/H ∩U ][H/H ∩U : K/K ∩U ] [G : H][H : K]. So [G : K] [G : H][H : K]. Given an open normal subgroup U of G and

an open normal subgroup V of H, let U 0 be an open normal subgroup of G such that H ∩ U 0 ⊂ V , and let U 00 = U ∩ U 0 . We have [G : HU ][H : KV ] [G : HU 00 ][H : K(H ∩ U 00 )]. But

[G : HU ][H : KV ] = [G/U : H/H ∩ U ][H/V : K/K ∩ V ], [G : HU ][H : K(H ∩ U 00 )] = [G/U 00 : H/H ∩ U 00 ][H/H ∩ U 00 : K/K ∩ U 00 ] = [G/U 00 : K/K ∩ U 00 ]. 00

It follows that

[G/U : H/H ∩ U ][H/V : K/K ∩ V ] [G/U 00 : K/K ∩ U 00 ], and hence [G : H][H : K] [G : K]. (ii) Use the fact that the set of open subgroups of G/K containing H/K are in one-to-one correspondence with the set of open subgroups of G containing H. (iii) Suppose H is an open subgroup of G. Then {gH}g∈G is an open covering of G. Since G is compact, we have G ⊂ g1 H ∪ · · · ∪ gn H for finitely many g1 , . . . , gn ∈ G. It follows that [G : H] is finite. Conversely, suppose [G : H] is finite. We can find an open subgroup V of G containing H such that [G : V ] = [G : H]. Suppose V 6= H. Take x ∈ V − H. We have 1 6∈ xH. As xH is closed, there exists an open normal subgroup U of G such that U ∩ xH = Ø. Then x 6∈ U H. Hence V ∩ U H is strictly contained in V . But V ∩ U H is an open subgroup containing H. So we have [G : H] ≥ [G : V ∩ U H] > [G : V ].

This contradicts [G : V ] = [G : H]. So we must have V = H. (iv) For any open subgroup V of G, using the fact that G−V is compact, one can show that ∩i Hi ⊂ V if and only if Hi ⊂ V for some i. It follows that [G : H] is the least common multiple of [G : V ] for open subgroups V containing some Hi . Hence [G : H] is the least common multiple of [G : Hi ].

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Fix a prime number p. A profinite group G is called a pro-p-group if [G : 1] is a power of p, or equivalently, G is the inverse limit of an inverse system of finite p-groups. Given a profinite group G, a closed subgroup H of G is called a Sylow p-subgroup if H is a pro-p-group and [G : H] is prime to p. Proposition 4.2.9. (i) Let G be a profinite group. Any pro-p-subgroup H of G is contained in a Sylow p-subgroup. In particular, Sylow p-subgroups exist. Any two Sylow p-subgroups are conjugate in G. (ii) Let G → G0 be an epimorphism of profinite groups. Then the image of a Sylow p-subgroup of G in G0 is a Sylow p-subgroup. Proof. (i) For any open normal subgroup U of G, let SU be the set of Sylow p-subgroups of G/U containing the p-subgroup H/H ∩ U . For any two normal subgroups V ⊂ U of G, the canonical homomorphism G/V → G/U induces a map SV → SU . Since SU is finite and nonempty for each open normal subgroup U , we have limU SU 6= Ø. Let (PU ) ∈ limU SU and let ←− ←− P = limU PU . Then P is a pro-p-subgroup of G containing H. [G : P ] is ←− the least common multiple of [G/U : PU ]. Since |G/U| |PU | is prime to p, [G : P ] is prime to p. So P is a Sylow p-subgroup. Let P and P 0 be Sylow p-subgroups of G. For any open normal subgroup U of G, let AU be the set of elements x ∈ G/U such that x−1 (P U/U )x = P 0 U/U . Here we regard P U/U and P 0 U/U as subgroups of G/U . As they are Sylow p-subgroups of G/U (confer the proof of (ii)), AU is nonempty. Moreover AU is finite. Hence limU AU 6= Ø. Let (xU ) ∈ limU AU and choose ←− ←− x ∈ G so that its image in G/U is xU for each U . Then x−1 P x = P 0 . (ii) Let N be the kernel of G → G0 . We have G0 ∼ = G/N . Let P be a Sylow p-subgroup of G. Its image in G0 is identified with the subgroup P N/N of G/N . We have [G/N : P N/N ] = [G : P N ] [G : P ]. So [G/N : P N/N ] is prime to p. On the other hand, we have [P N/N : 1] = [P/P ∩ N : 1] = [P : P ∩ N ] [P : 1].

So [P N/N : 1] is a power of p. Therefore P N/N is a Sylow p-subgroup.

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Cohomology of Profinite Groups

([Serre (1964)] I 2.) Let G be a profinite group and let A be an abelian group on which G acts on the left. Put the discrete topology on A. Then the following conditions are equivalent: (i) G acts continuously on A, that is, the map G × A → A,

is continuous. (ii) For any x ∈ A, the stabilizer

(g, x) 7→ gx

Gx = {g ∈ G|gx = x}

is open in G. S (iii) We have A = U AU , where U goes over the set of open subgroups of G. From now on, we simply call a discrete abelian group on which G acts left continuously a G-module. For any G-module A, and any nonnegative integer n, let C n (G, A) be the set of continuous maps from Gn to A. Note that C n (G, −) is an exact functor on the category of G-modules. Define dn : C n (G, A) → C n+1 (G, A)

by

(dn f )(g1 , . . . , gn+1 ) = g1 f (g2 , . . . , gn+1 ) +

n X

(−1)j f (g1 , . . . , gj−1 , gj gj+1 , gj+2 , . . . , gn+1 )

j=1

n+1

+(−1)

f (g1 , . . . , gn )

n

for any f ∈ C (G, A) and (g1 , . . . , gn+1 ) ∈ Gn+1 . One can verify d ◦ d = 0. We define H n (G, A) to be the n-th cohomology group of the complex C · (G, A). We also denote H n (G, A) by Rn ΓG (A). Theorem 4.3.1. Let {Gi }i∈I be an inverse system of profinite groups, and let {Ai }i∈I be a direct system of abelian groups such that each Ai is a Gi module. For each pair i ≤ j, suppose that the homomorphism Ai → Aj is compatible with the homomorphism Gj → Gi and the group actions of Gi and Gj on Ai and Aj , respectively. Set G = limi Gi and A = limi Ai . Then ←− −→ the canonical chain map lim C · (Gi , Ai ) → C · (G, A) −→ i

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is an isomorphism, and hence H · (G, A) ∼ H · (Gi , Ai ). = lim −→ i

Gni

Proof. Let f1 , f2 : → Ai be two continuous maps. Assume they induce the same map from Gn to A. Let f10 , f20 : Gni → A be the maps induced by f1 and f2 , respectively, and let X = {x ∈ Gni |f10 (x) = f20 (x)}.

Since A has the discrete topology and f10 and f20 are continuous, X is open in Gni . Let πi : Gn = lim Gni → Gni , ←−

πji : Gnj → Gni

i

(j ≥ i)

be the canonical homomorphisms. Note that \ im πi = im πji .

Indeed, given any x ∈

T

Aj1 j2 = {(xj ) ∈

j≥i

j≥i

Y j≥i

im πji and any j1 ≥ j2 ≥ i, let

Gnj |πj1 j2 (xj1 ) = xj2 and πj2 i (xj2 ) = x}.

Q Then Aj1 j2 are closed in j≥i Gnj , and any intersection of finitely many of Q T them is nonempty. Since j≥i Gnj is compact, we have j1 ≥j2 ≥i Aj1 j2 6= Ø. T For any y ∈ j1 ≥j2 ≥i Aj1 j2 , we have y ∈ Gn = limi Gni and πi (y) = x. ←− So x ∈ im πi . By our assumption, we have im πi ⊂ X. It follows that T n n j≥i im πji ⊂ X. As Gi − X is compact and Gi − im πji (j ≥ i) form n an open covering of Gi − X, we have im πji ⊂ X for a sufficiently large j. Then f10 ◦ πji = f20 ◦ πji . Since Gnj is compact and A is discrete, im(f10 ◦ πji ) = im(f20 ◦ πji ) is finite. Taking j sufficiently large, we see that f1 and f2 induce the same map from Gnj to Aj . This proves the canonical map limi C n (Gi , Ai ) → C n (G, A) is injective. −→ Given a continuous map f : Gn → A, let im f = {a1 , . . . , ak }. For each aλ , f −1 (aλ ) is open and closed. So each f −1 (aλ ) is a finite union of sets of the form ! ! Y \ Ui lim Gni , ←− i

i

Gni

Gni

where Ui ⊂ are open and Ui = for all but finitely many i. Taking j sufficiently large so that for any i ≥ j, all Ui appeared above are Gni . Then for any x, x0 ∈ Gn such that πj (x) = πj (x0 ), we have f (x) = f (x0 ). So f can be factorized as a composite πj

f0

Gn → πj (Gn ) → A

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for some continuous map f 0 : πj (Gn ) → A. We can find disjoint open and closed subsets Vλ of Gnj such that f 0−1 (aλ ) ⊂ Vλ . Taking j sufficiently large, we may assume that there exist aλj ∈ Aj with images aλ in A, respectively. Define fj : Gnj → Aj by fj (Vλ ) = {aλj } and fj (Gnj − ∪kλ=1 Vλ ) = {0}. Then fj is continuous and induces f . This proves the canonical map limi C n (Gi , Ai ) → C n (G, A) is surjective. −→ Corollary 4.3.2. Let G be a profinite group and let A be a G-module. Then we have H i (G, A) ∼ H i (G/U, AU ), = lim −→ U

where U goes over the set of open normal subgroups of G. For all i ≥ 1, H i (G, A) are torsion groups. Proof.

Use 4.3.1 and 4.1.6.

Corollary 4.3.3. For each i, H i (G, −) is the i-th derived functor of the functor A 7→ AG on the abelian category of G-modules. Proof. One can check H 0 (G, A) = AG . Using the fact that C i (G, −) are exact functors, one can show that H i (G, −) form a δ-functor. The abelian category of G-modules has enough injective objects. For any injective Gmodule I, one can check that I U is an injective G/U -module for any open normal subgroup U of G. It follows that H i (G, I) = lim H i (G/U, I U ) = 0 −→ U

for any i ≥ 1. Our assertion follows.

Let f : G0 → G be a continuous homomorphism of profinite groups, A a G-module, A0 a G0 -module, φ : A → A0 an additive map compatible with 0 the group actions. Then φ induces a homomorphism AG → A0G . It extends canonically to a family of homomorphisms H i (G, A) → H i (G0 , A0 ). These homomorphisms are induced by the chain map C · (G, A) → C · (G0 , A0 ) induced by f and φ. In particular, if H is a closed subgroup of G and A is a G-module, then we have the restriction homomorphisms Res : H i (G, A) → H i (H, A). Let H be a closed subgroup of G, let B be an H-module, and let IndG HB be the group of continuous maps φ : G → B satisfying φ(hg) = hφ(g) for

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0 0 any g ∈ G and h ∈ H. Define a G-action on IndG H B by (gφ)(g ) = φ(g g) G for any φ ∈ IndH B and g, g 0 ∈ G. Note that this action is continuous if we put the discrete topology on IndG H B. Let

θ : IndG HB → B

be the homomorphism defined by θ(φ) = φ(1) for any φ ∈ IndG H B. It is compatible with the group actions. We again have the following theorem: Theorem 4.3.4 (Shapiro). For any H-module B, θ induces isomor∼ = i phisms H i (G, IndG H B) → H (H, B) for all i. Consider the case H = {1}. Any abelian group M can be considered as an H-module. We can form the G-module IndG {1} M of continuous maps from G to M . A G-module is called induced if it is isomorphic to IndG {1} M for some abelian group M . A G-module is called weakly injective if it is a direct factor of an induced G-module. For any weakly injective G-module I, we have H i (G, I) = 0 for any i ≥ 1. Any G-module A can be embedded into an induced G-module. Injective G-modules are weakly injective. We can use resolutions of A by weakly injective G-modules to calculate H i (G, A). Let H be a closed subgroup of G. Then for any abelian group M , we have H i (H, IndG {1} M ) = 0 for any i ≥ 1. Indeed, we have U H i (H, IndG H i (H/H ∩ U, (IndG {1} M ) = lim {1} M ) ), −→ U

where U goes over the set of open normal subgroups of G. We have (IndG M )U ∼ = Hom(Z[G/U ], M ). {1}

U So (IndG {1} M ) is an induced G/U -module and thus an induced (H/H ∩U )U module. It follows that H i (H/H ∩ U, (IndG {1} M ) ) = 0 and hence G i H (H, Ind{1} M ) = 0 for any i ≥ 1. For any weakly injective G-module A, we also have H i (H, A) = 0 for any i ≥ 1. Using this fact, we can show that if H is a normal closed subgroup of G and A is a G-module, then we have a biregular spectral sequence

E2ij = H i (G/H, H j (H, A)) ⇒ H i+j (G, A),

which we call the Hochschild–Serre spectral sequence. For any open subgroup H of G and any G-module A, we can define the corestriction homomorphisms Cor : H i (H, A) → H i (G, A). Proposition 4.3.5. Let H be an open subgroup of G with index n. Then Cor ◦ Res = n.

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Proposition 4.3.6. Let H be a closed subgroup of a profinite group G such that [G : H] is relatively prime to a prime number p. Then for any G-module A and any i, the restriction homomorphism Res : H i (G, A) → H i (H, A) is injective on the p-primary part of H i (G, A). Proof. When [G : H] < ∞, this follows from 4.3.5. In general, we have H i (H, A) = limU H i (U, A) by 4.3.1, where U goes over the set of open −→ subgroups of G containing H. For each U , [G : U ] is prime to p. By 4.3.5, Res : H i (G, A) → H i (U, A) is injective on the p-primary part of H i (G, A). It follows that Res : H i (G, A) → H i (H, A) is injective on the p-primary part of H i (G, A). ˆ = lim Proposition 4.3.7. Let Z Z/n, where the partial order on the ←−n∈N direct set N is defined by n ≤ m if and only if n m and the transition homomorphism Z/m → Z/n is the canonical one. For any torsion abelian ˆ action, we have group A with continuous Z  ˆ   AZ if i = 0, i ˆ H (Z, A) ∼ = AZˆ if i = 1,  0 if i 6= 0, 1.

Proof. Let T = (¯ 1) ∈ limn∈N Z/n be the canonical topological generator ←− ˆ of Z. For any n ∈ N, let n

AT = {x ∈ A|T n (x) = x}.

We have n

ˆ A) ∼ H i (Z, H i (Z/n, AT ). = lim −→ n For any m, n ∈ N, consider the following commutative diagram: n

T −1

n

T n−1 +···+T +1

mn

T −1

mn

T mn−1 +···+T +1

0 → AT ↓1

0 → AT

→ AT ↓1

→ AT

T n−1 +···+T +1

→

T mn−1 +···+T +1

→

→

→

T −1

n

AT

mn

T −1

mn

→ AT → ↓m

→ AT

n

T −1

n

T n−1 +···+T +1

mn

T −1

mn

T mn−1 +···+T +1

→ AT AT 2 ↓m ↓ m2

AT

n

AT ↓m

→ AT

→

→

→

n

AT → · · · ↓ m3

AT

mn

→ ··· ,

where the vertical arrows are multiplication by powers of m. By the proof of 4.1.2, the cohomology groups of the two horizontal complexes are

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mn

H · (Z/n, AT ) and H · (Z/mn, AT ), respectively. Moreover, the vertical arrows in the above commutative diagram induce the transition homomorn ˆ A) ∼ phisms for the direct limit H i (Z, H i (Z/n, AT ). = lim −→n We have ˆ A) ∼ H 0 (Z, ker(T − 1|AT n ), = lim −→ n

where the transition homomorphism of the direct system are inclusions ker(T − 1|AT n ) ,→ ker(T − 1|AT mn ). It follows that ˆ

ˆ A) ∼ H 0 (Z, = AZ . ˆ A). Of course, this also follows from the definition of H 0 (Z, Set A0 = {x ∈ A|T n (x) = x and (T n−1 + · · · + T + 1)(x) = 0 for some n ∈ N}. S n Using the fact that A = n∈N AT , we see that n ˆ A) ∼ H 1 (Z, H 1 (Z/n, AT ) = lim −→

n

∼ ker(T n−1 + · · · + T + 1|AT n )/im(T − 1|AT n ) = lim −→ n

∼ = A0 /(T − 1)A.

ˆ Since A is a torsion Z-module, for any x ∈ A, there exist m, n ∈ N such Tn that mx = 0 and x ∈ A . We then have (T mn−1 + · · · + T + 1)(x) = m(T n−1 + · · · + T + 1)(x) = 0. Hence x ∈ A0 . Therefore A = A0 and ˆ A) ∼ H 1 (Z, = A0 /(T − 1)A = AZˆ . Using the fact that A is a torsion abelian group, one sees that for any i ≥ n 2 and any x ∈ H i (Z/n, AT ), there exists m ∈ N such that the transition homomorphism n

H i (Z/n, AT ) → H i (Z/mn, AT n

mn

)

ˆ A) = lim H i (Z/n, AT ) maps x to 0. It follows for the direct limit H i (Z, −→n ˆ A) = 0 for any i ≥ 2. that H i (Z,

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Fix a prime number p. Put a topology on Z so that subgroups nZ with n relatively prime to p form a base of neighborhoods of 0. The completion ˆ (p) . We have of Z with respect to this topology is denoted by Z ˆ (p) = lim Z/n. Z ←− (n,p)=1

Let q be a power of p. Note that multiplication by q induces a continuous isomorphism of groups ˆ (p) → Z ˆ (p) . q:Z ˆ (p) -module. Suppose that every element Proposition 4.3.8. Let A be a Z in A is annihilated by some integer relatively prime to p. Then we have  (p) ˆ  if i = 0,  AZ i ˆ (p) ∼ H (Z , A) = AZˆ (p) if i = 1,  0 if i 6= 0, 1.

Let φ : A → A be a homomorphism compatible with the homomorphism ˆ (p) → Z ˆ (p) and the group actions, that is, q:Z φ((qg)x) = gφ(x)

ˆ (p) and x ∈ A. Then the homomorphism induced by (q, φ) on for any g ∈ Z ˆ (p) 0 ˆ (p) H (Z , A) ∼ = AZ is the homomorphism ˆ (p)

φ : AZ

ˆ (p)

→ AZ

ˆ (p) , A) ∼ induced by φ, and the homomorphism induced by (q, φ) on H 1 (Z = AZˆ (p) is the homomorphism qφ : AZˆ (p) → AZˆ (p) induced by qφ. ˆ (p) , A) can be proved Proof. The statement about the structure of H i (Z by the same method as in the proof of 4.3.7. The determination of the ˆ (p) , A) can be done using the homomorphism induced by (q, φ) on H 0 (Z 0 ˆ (p) definition of H (Z , A). ˆ (p) , A), we To determine the homomorphism induced by (q, φ) on H 1 (Z 1 ˆ (p) first calculate H (Z , A) using its definition. A 1-cocycle is a continuous ˆ (p) → A satisfying map f : Z f (g1 + g2 ) = g1 f (g2 ) + f (g1 )

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ˆ (p) . Let T = (¯1) ∈ Z ˆ (p) = lim for any g1 , g2 ∈ Z Z/n be the canonical ←−(n,p)=1 ˆ (p) . Then a 1-cocycle f is completely determined topological generator of Z by the value f (T ) ∈ A. Conversely, given a ∈ A, define f : Z → A by f (0) = 0, f (1) = a, f (n) = T n−1 f (1) + f (n − 1),

f (−n) = −T −n f (n) for any n ∈ N. Note that we have f (g1 + g2 ) = g1 f (g2 ) + f (g1 ) for any g1 , g2 ∈ Z. We claim that f : Z → A is continuous. Indeed, we can find m, n ∈ N which are relatively prime to p such that ma = 0 and n a ∈ AT . We then have (T mn−1 + · · · + T + 1)(a) = m(T n−1 + · · · T + 1)(a) = 0. It follows that f (mn) = 0. This implies that f (δ) = 0 for any δ ∈ mnZ. For any g ∈ Z and any δ ∈ mnZ, we then have f (g + δ) = f (g). ˆ (p) → A. Thus f is continuous. We can then extend f to a 1-cocycle f : Z Therefore the group of 1-cocycles can be identified with A. A 1-cocycle ˆ (p) → A is a 1-coboundary if there exists a ∈ A such that f (g) = ga−a f :Z ˆ (p) . Under the above identification, the group of 1-coboundaries for all g ∈ Z is identified with the subgroup {T a − a|a ∈ A} of A. We thus have ˆ (p) , A) ∼ H 1 (Z = A/{T a − a|a ∈ A} = AZˆ (p) . Note that φ induces a homomorphism on AZˆ . To see this, it suffices to ˆ (p) , x ∈ A} is invariant under φ. Given show that the subset {gx − x|g ∈ Z ˆ (p) , there exists g 0 ∈ Z ˆ (p) such that qg 0 = g. We then have g∈Z φ(gx − x) = φ((qg 0 )x) − φ(x) = g 0 φ(x) − φ(x). ˆ (p) → A be a 1-cocyle and let a = f (T ). This proves our assertion. Let f : Z 1 ˆ (p) The homomorphism on H (Z , A) induced by (φ, q) maps the cohomology class of f to the cohomology class of φ ◦ f ◦ q. We have ! q−1 X q−1 i φ ◦ f ◦ q(T ) = φ(T (a) + · · · + T (a) + a) = qφ(a) + φ (T (a) − a) . As φ

q−1 P i=0

i=0 (T i (a) − a) lies in the subgroup generated by gx − x (g ∈

ˆ (p) , x ∈ A), φ ◦ f ◦ q(T ) has the same image in A ˆ (p) as qφ(a). ThereZ Z ˆ (p) , A) ∼ fore the homomorphism induced by (q, φ) on H 1 (Z = AZˆ (p) is the homomorphism qφ : AZˆ (p) → AZˆ (p) induced by qφ.

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Proposition 4.3.9. Let k be a separably closed field of characteristic p. For any positive integer n relatively prime to p, let µn (k) = {ζ ∈ k|ζ n = 1}

be the group of n-th roots of unity in k, let G = lim(n,p)=1 µn (k), and let A ←− be a G-module such that every element in A is annihilated by some integer relatively prime to p. Then we have  G  if i = 0, A i ∼ µ , A ) if i = 1, H (G, A) = Hom(lim ←−(n,p)=1 n G  0 if i 6= 0, 1.

Let q be a power of p, let q : G → G be the isomorphism of raising to q-th power, and let φ : A → A be a homomorphism compatible with the homomorphism q : G → G and the group actions, that is, φ((qg)x) = gφ(x)

for any g ∈ G and x ∈ A. Then the homomorphism induced by (q, φ) on H 0 (G, A) ∼ = AG is the homomorphism φ : AG → AG induced by φ, and the homomorphism induced by (q, φ) on H 1 (G, A) ∼ µ ,A ) = Hom(lim ←−(n,p)=1 n G is the homomorphism Hom( lim µn , AG ) → Hom( lim µn , AG ) ←− ←− (n,p)=1

(n,p)=1

that maps any ψ ∈ Hom(lim(n,p)=1 µn , AG ) to φ ◦ ψ ◦ q. ←−

Proof. For each positive integer n relatively prime to p, choose a primn itive n-th root of unity ζn in k such that ζnm = ζm whenever m|n. Then ζ = (ζn ) is a topological generator of G = lim(n,p)=1 µn (k). We have an ←− isomorphism ∼ lim Z/n lim µn (k) = ←− ←− (n,p)=1

(n,p)=1

that maps ζ to the canonical topological generator 1 = (¯1) of lim(n,p)=1 Z/n. ←− By 4.3.8, we have  G if i = 0, A H i (G, A) ∼ = AG if i = 1,  0 if i 6= 0, 1. ∼ AG By the proof of 4.3.8, in the case i = 1, the isomorphism H 1 (G, A) =

is induced by the map that maps each 1-cocycle f : G → A to f (ζ). This isomorphism depends on the choice of ζ. Consider the map H 1 (G, A) ⊗Z(p) ( lim µn (k)) → AG ←− (n,p)=1

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defined by [f ] ⊗ g → f (g), where [f ] denotes the cohomology class of any 1-cocycle f , and g is any element in G = lim(n,p)=1 µn (k). One can show that it is a well-defined ho←− momorphism. Through the isomorphism lim(n,p)=1 µn (k) ∼ Z/n = lim ←− ←−(n,p)=1 defined above, this homomorphism is identified with the isomorphism H 1 (G, A) ∼ = AG defined in the proof of 4.3.8. So we have H 1 (G, A) ⊗Z(p) ( lim µn (k)) ∼ = AG , ←− (n,p)=1

and hence we have a canonical isomorphism H 1 (G, A) ∼ lim µn (k), AG ). = Hom( ← − (n,p)=1

The other statements of the proposition follows directly from 4.3.8. 4.4

Cohomological Dimensions

([Serre (1964)] I 3.) Let G be a profinite group. For each prime number p, we define the pcohomological dimension cdp (G) (resp. the strict p-cohomological dimension scdp (G)) of G to be the smallest integer n such that for any torsion G-module (resp. any G-module) A and any i > n, the p-primary part of H i (G, A) vanishes. We define the cohomological dimension cd(G) (resp. the strict cohomological dimension scd(G)) to be supp (cdp (G)) (resp. supp (scdp (G))). Proposition 4.4.1. Let G be a profinite group. The following conditions are equivalent: (i) cdp (G) ≤ n. (ii) H i (G, A) = 0 for any i > n and any p-torsion G-module A. (iii) H n+1 (G, A) = 0 for any simple p-torsion G-module A. L Proof. For any torsion G-module A, we have A = p A(p), where for each prime p, A(p) is the p-primary part of A. Note that H i (G, A(p)) is the p-primary part of H i (G, A) for each i. So (i)⇔(ii). (i)⇒(iii) is clear. Suppose that (iii) holds. Any finite p-torsion G-module has a finite filtration with simple successive quotients, and any torsion G-module is the direct

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limit of its finite G-submodules. So H n+1 (G, A) = 0 for any p-torsion Gmodule A. A can be embedded into the induced G-module IndG {1} A. Since G i i+1 i ∼ H (G, Ind{1} A) = 0 for all i ≥ 1, we have H (G, A) = H (G, IndG {1} A/A) for any i ≥ 1. IndG A/A is also a p-torsion G-module. By induction on i, {1} i we see that H (G, A) = 0 for any i > n and any p-torsion G-module A. So (ii) holds. Proposition 4.4.2. For any profinite group G, we have cdp (G) ≤ scdp (G) ≤ cdp (G) + 1. Proof. The first inequality follows from the definition of cdp (G) and scdp (G). For any G-module A, let Ap = ker(p : A → A). We have exact sequences p

0 → Ap → A → pA → 0,

0 → pA → A → A/pA → 0.

Note that Ap and A/pA are p-torsion G-modules. So H i (G, Ap ) and H i−1 (G, A/pA) vanish for any i > cdp (G) + 1. Hence p

H i (G, A) → H i (G, pA),

H i (G, pA) → H i (G, A)

are injective for any i > cdp (G) + 1. It follows that their composite p

H i (G, A) → H i (G, A)

is injective. This implies that the p-primary part of H i (G, A) vanishes for any i > cdp (G) + 1. So scdp (G) ≤ cdp (G) + 1. Proposition 4.4.3. Let H be a closed subgroup of a profinite group G. Then cdp (H) ≤ cdp (G),

scdp (H) ≤ scdp (G).

If [G : H] is relatively prime to p, then cdp (H) = cdp (G),

scdp (H) = scdp (G).

Proof. The first statement follows from 4.3.4. The second statement follows from 4.3.6. Corollary 4.4.4. Let Gp be a Sylow p-subgroup of a profinite group G. Then cdp (G) = cdp (Gp ) = cd(Gp ), scdp (G) = scdp (Gp ) = scd(Gp ).

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Proof. By 4.4.3, we have cdp (G) = cdp (Gp ) and scdp (G) = scdp (Gp ). Moreover, for any prime number ` distinct form p, we have cd` (Gp ) = cd` ({1}) = 0, scd` (Gp ) = scd` ({1}) = 0. It follows that cdp (Gp ) = cd(Gp ) and scdp (Gp ) = scd(Gp ).

Proposition 4.4.5. Let H be a closed normal subgroup of G. We have cdp (G) ≤ cdp (H) + cdp (G/H). Proof.

Use the Hochschild–Serre spectral sequence.

Lemma 4.4.6. Let G be a finite p-group and let A be a nonzero p-torsion G-module. Then AG 6= 0. Proof. Replacing A by a subgroup generated by an orbit of a nonzero element of A, we may assume that A is finite. Since A is a nonzero ptorsion abelian group, we have |A| = pn for some positive integer n. Take x1 , . . . , xn so that A is the disjoint union of the orbits Gx1 , . . . , Gxn . For any xi , we have a bijection G/Gxi → Gxi ,

gGxi 7→ gxi ,

where Gxi is the stabilizer of xi . Hence |Gxi | = |G/Gxi |. Since G is a finite p-group, we have |G/Gxi | = pni for some nonnegative integer ni , and we have ni = 0 if and only if xi ∈ AG . So we have k X X pn = |A| = |Gxi | = pni + |AG |. i=1

n ≥1

i G G This implies that p |A |. But |A | 6= 0 since 0 ∈ AG . So AG contains at least p elements.

Lemma 4.4.7. Let G be a pro-p-group. Any nonzero simple p-torsion Gmodule is isomorphic to Z/p with the trivial G-action.

Proof. Let A be a nonzero simple p-torsion G-module. It is necessarily finite. The action of G on A factors through a finite quotient group. By 4.4.6, we have AG 6= 0. Since A is simple, we must have A = AG . So G acts trivially on A. A has no nontrivial subgroup. So we have A = Z/p. Proposition 4.4.8. Let G be a pro-p-group. Then cd(G) ≤ n if and only if H n+1 (G, Z/p) = 0. Proof. 4.4.7.

By 4.4.4, we have cd(G) = cdp (G). We then apply 4.4.1 (iii) and

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Etale Cohomology Theory

Galois Cohomology

([Serre (1964)] II 1–4, [SGA 4 21 ] Arcata III 2.) In this section, K is a field and we fix a separable closure Ks of K. We study the cohomology of the profinite group Gal(Ks /K) = lim Gal(L/K), ←− L

where L goes over the family of finite galois extensions of K in Ks . For any Gal(Ks /K)-module A, by 4.3.2, we have H i (Gal(Ks /K), A) = lim H i (Gal(L/K), AGal(Ks /L) ). −→ L

Lemma 4.5.1. Let L be an extension of K and let σ1 , . . . , σn be a finite family of distinct K-automorphisms of L. Then {σ1 , . . . , σn } is linearly independent over L, that is, if {a1 , . . . , an } is a family of elements in L n P such that ai σi (x) = 0 for all x ∈ L, then ai = 0 for all i. i=1

Proof. Suppose {σ1 , . . . , σn } is not linearly independent over L. Let m be the smallest positive integer such that there exist distinct i1 , . . . , im ∈ {1, . . . , n} and nonzero a1 , . . . , am ∈ L − {0} satisfying a1 σi1 + · · · + am σim = 0. Note that we must have m ≥ 2. For any x, y ∈ L, we have a1 σi1 (xy) + · · · + am σim (xy) = 0,

that is, a1 σi1 (x)σi1 + · · · + am σim (x)σim = 0. From these equations, we get a2 (σi1 (x) − σi2 (x))σi2 + · · · + am (σi1 (x) − σim (x))σim = 0. Since σi1 6= σi2 , there exists x ∈ L such that σi1 (x) 6= σi2 (x). But then the last equation above contradicts the minimality of m. So {σ1 , . . . , σn } is linearly independent over L. Lemma 4.5.2. Let L/K be a finite galois extension and let Gal(L/K) = {σ1 , . . . , σn }. Then there exists x ∈ L such that {σ1 (x), . . . , σn (x)} is a basis of L over K.

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A basis of L over K of the above form is called a normal basis. Proof. Note that {σ1 (x), . . . , σn (x)} is a basis if and only if P det(σi σj (x)) 6= 0. Indeed, suppose that det(σi σj (x)) 6= 0 and j aj σj (x) = P 0 for some aj ∈ K. Then j aj σi σj (x) = 0 for all i. This implies that aj = 0. Hence {σ1 (x), . . . , σn (x)} is linearly independent. As n = |Gal(L/K)| = [L : K], {σ1 (x), . . . , σn (x)} is a basis of L over K. Suppose det(σi σj (x)) = 0. Then there exist b1 , . . . , bn ∈ L, not all equal to P 0, such that j bj σi σj (x) = 0 for all i. Without loss of generality, assume b1 6= 0. By 3.2.2, there exists b ∈ L such that TrL/K (b) 6= 0. Replacing bj P by bb−1 TrL/K (b1 ) 6= 0. We have j σi−1 (bj )σj (x) = 0. 1 bj , we may assume P Summing over i, we get j TrL/K (bj )σj (x) = 0. So {σ1 (x), . . . , σn (x)} is not linearly independent over K. Assume K is infinite and let us prove the existence of a normal basis. Define p(i, j) ∈ {1, . . . , n} by σi σj = σp(i,j) . Consider the polynomial f (X1 , . . . , Xn ) = det(Xp(i,j) ). We claim f (X1 , . . . , Xn ) 6= 0. Indeed, if p(i, j) = p(i0 , j) for some i, i0 , j, then i = i0 ; if p(i, j) = p(i, j 0 ) for some i, j, j 0 , then j = j 0 . So f (1, 0, . . . , 0) is the determinant of a matrix with only one nonzero entry on each row and each column. We have f (1, 0, . . . , 0) = ±1 and hence f (X1 , . . . , Xn ) 6= 0. We have det(σi σj (x)) = f (σ1 (x), . . . , σn (x)). Let {x1 , . . . , xn } be a basis of L over K. Any x ∈ L can be written as P P x = j aj xj (aj ∈ K). We have σi (x) = j aj σi (xj ). Let X X g(Y1 , . . . , Yn ) = f σ1 (xj )Yj , . . . , σn (xj )Yj . j

j

P We claim that (σi (xj )) is an invertible matrix. Indeed, if i bi σi (xj ) = 0 P for some b1 , . . . , bn ∈ L and all j, then i bi σi = 0. By 4.5.1, we have bi = 0 for all i. So (σi (xj )) is an invertible matrix. Let (bij ) be its inverse. Then X X f (X1 , . . . , Xn ) = g b1j Xj , . . . , bnj Xj . j

j

As f (X1 , . . . , Xn ) 6= 0, we have g(Y1 , . . . , Yn ) 6= 0. Since K is an infinite field, there exist a1 , . . . , an ∈ K such that g(a1 , . . . , an ) 6= 0, that is, X X f σ1 aj xj , . . . , σn aj xj 6= 0. j

j

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P Let x = j aj xj . Then det(σi σj (x)) 6= 0. Hence {σ1 (x), . . . , σn (x)} is a normal basis. Assume Gal(L/K) is cyclic. (This holds if K is finite.) Let us prove the existence of a normal basis. Choose a generator σ of Gal(L/K). Define a K[T ]-module structure on L by X X ai σ i (x) ai T i · x = P

i

i

i

for any i ai T ∈ K[T ] and x ∈ L. Note that σ satisfies the polynomial equation T n − 1 = 0. By 4.5.1, σ does not satisfy any polynomial equation of degree < n. So T n − 1 is the minimal polynomial of σ. Since [L : K] = n, we have L ∼ = K[T ]/(T n − 1) as K[T ]-modules. (We leave the proof of this fact to the reader as an exercise in linear algebra.) Let x be the element in L corresponding to 1 in K[T ]/(T n − 1). Then {x, σ(x), . . . , σn−1 (x)} is a normal basis. Theorem 4.5.3. For any finite galois extension L/K, we have K if i = 0, i H (Gal(L/K), L) = 0 if i ≥ 1, and hence

i

H (Gal(Ks /K), Ks ) =

K 0

if i = 0, if i ≥ 1.

Proof. By 4.5.2, L is an induced Gal(L/K)-module. The first statement follows. The second statement then follows from the fact that H i (Gal(Ks /K), Ks ) ∼ H i (Gal(L/K), L), = lim −→ L

where L goes over the family of finite galois extensions of K contained in Ks . Theorem 4.5.4 (Hilbert 90). For any finite galois extension L/K, we have H 1 (Gal(L/K), L∗ ) = 0, and hence H 1 (Gal(Ks /K), Ks∗ ) = 0. Proof. Let f : Gal(L/K) → L∗ be a 1-cocycle. By Theorem 4.5.1, there exists x ∈ L such that X f (σ)σ(x) 6= 0. Let y =

P

σ∈Gal(L/K)

σ∈Gal(L/K)

f (σ)σ(x) ∈ L∗ . For any τ ∈ Gal(L/K), we have f (τ σ) = τ (f (σ)) · f (τ ),

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and τ (y) =

X

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τ (f (σ))τ σ(x)

σ∈Gal(L/K)

=

X

f (τ )−1 f (τ σ)τ σ(x)

σ∈Gal(L/K)

= f (τ )−1 y. So we have f (τ ) = yτ (y)−1 . This shows that f is a 1-coboundary. Therefore H 1 (Gal(Ks /K), Ks∗ ) = 0. In 5.7.9, we will give another proof of the Hilbert 90 using the descent theory and the etale cohomology. Theorem 4.5.5. (i) If K is a field of characteristic p, then cdp (Gal(Ks /K)) ≤ 1. (ii) Assume H 2 (Gal(Ks /L), Ks∗ ) = 0 for any finite extension L of K contained in Ks . Then cd(Gal(Ks /K)) ≤ 1, that is, for any torsion Gal(Ks /K)-module A, we have H i (Gal(Ks /K), A) = 0 for any i ≥ 2. Moreover, we have

H i (Gal(Ks /K), Ks∗) = 0

for any i ≥ 1. Proof. G (i) Let Gp be the Sylow p-subgroup of Gal(Ks /K) and let M = Ks p . We have an exact sequence of Gal(Ks /K)-modules ℘

0 → Z/p → Ks → Ks → 0,

where ℘(x) = xp − x for any x ∈ Ks . By 4.5.3, we have

H i (Gp , Ks ) = H i (Gal(Ks /M ), Ks ) = 0

for any i ≥ 1. The long exact sequence of cohomology groups associated to the above short exact sequence shows that H i (Gp , Z/p) = 0 for any i ≥ 2. By 4.4.8, we have cd(Gp ) ≤ 1. By 4.4.4, we have cdp (Gal(Ks /K)) ≤ 1. (ii) Let ` be a prime number distinct from the characteristic of K, let G` be the Sylow `-subgroup of Gal(Ks /K), and let M = KsG` . We have M = limL L, where L goes over the set of finite extensions of K contained −→ in M . We have an exact sequence of Gal(Ks /K)-modules x7→x`

0 → µ` (Ks ) → Ks∗ → Ks∗ → 0,

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where µ` (Ks ) is the group of `-th roots of unity in Ks∗ . For any finite extension L of K contained in M , we have H i (Gal(Ks /L), Ks∗) = 0 for i = 1, 2 by 4.5.4 and the assumption. The long exact sequence of cohomology groups associated to the above short exact sequence then shows that H 2 (Gal(Ks /L), µ` (Ks )) = 0. By 4.3.1, we have H 2 (G` , µ` (Ks )) = lim H 2 (Gal(Ks /L), µ` (Ks )) = 0. −→ L

Note that µ` (Ks ) has no nontrivial subgroup. So as a G` -module, it is simple and isomorphic to Z/` with the trivial G` -action. By 4.4.8, we have cd(G` ) ≤ 1. By 4.4.4, we have cd` (Gal(Ks /K)) ≤ 1. Combined with (i), we get cd(Gal(Ks /K)) ≤ 1. By 4.4.2, we have scd(Gal(Ks /K)) ≤ 2. It follows that H i (Gal(Ks /K), Ks∗) = 0 for any i ≥ 3.

A field K is called quasi-algebraically closed if for any positive integer n and any n-variable homogeneous polynomial f (X1 , . . . , Xn ) of degree < n with coefficients in K, f (X1 , . . . , Xn ) = 0 has nonzero solutions in K n . In 5.7.15, we will prove the following using the descent theory and the etale cohomology: Proposition 4.5.6. Suppose K is quasi-algebraically closed. H 2 (Gal(Ks /K), Ks∗ ) = 0.

We have

Lemma 4.5.7. If K is quasi-algebraically closed and L is algebraic over K, then L is quasi-algebraically closed. Proof. Let F (X1 , . . . , Xn ) be an n-variable homogeneous polynomial of degree < n with coefficients in L. Then the coefficients of F (X1 , . . . , Xn ) lies in a finite extension of K contained in L. So to prove that F (X1 , . . . , Xn ) = 0 has a nonzero solution, we may assume that [L : K] < ∞. Let {e1 , . . . , em } be a basis of L over K. Consider the function f (x11 , . . . , x1m , . . . , xn1 , . . . , xnm )

= NormL/K (F (x11 e1 + · · · + x1m em , . . . , xn1 e1 + · · · + xnm em ))

defined on K nm . It is a homogeneous polynomial of degree < nm with coefficients in K. Since K is quasi-algebraically closed, f (x11 , . . . , x1m , . . . , xn1 , . . . , xnm ) has a nonzero solution in K nm . Hence F (X1 , . . . , Xn ) has a nonzero solution in Ln .

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Theorem 4.5.8 (Tsen). Let k be an algebraically closed field and let K/k be an extension of transcendental degree 1. Then K is quasi-algebraically closed. Proof. By 4.5.7, it suffices to consider the case where K = k(t) is purely transcendental. Let X f (X1 , . . . , Xn ) = ai1 ...in (t)X1i1 · · · Xnin i1 +···+in =d

be an n-variable homogeneous polynomial of degree d < n with coefficients ai1 ...in (t) ∈ k(t). We need to show that f (X1 , . . . , Xn ) has a nonzero solution in k(t)n . It suffices to treat the case where ai1 ...in (t) are polynomials. Let δ = max{deg(ai1 ...in (t))|i1 + · · · + in = d}. P j Substituting Xi = N j=0 aij t (i = 1, . . . , n) into f (X1 , . . . , Xn ), and setting the coefficients of tk (k = 0, 1, . . . , δ + dN ) to zero, we get a system of δ + dN + 1 homogeneous equations with n(N + 1) unknowns aij . Since d < n, we have δ + dN + 1 < n(N + 1) for sufficiently large N . In this case, the system of homogeneous equations has a nonzero solution by [Hartshorne (1977)] I 7.2. So f (X1 , . . . , Xn ) = 0 has a nonzero solution. Theorem 4.5.9. Let K be a quasi-algebraically closed field. Then for any torsion Gal(Ks /K)-module A, we have H i (Gal(Ks /K), A) = 0 for any i ≥ 2. Moreover we have H i (Gal(Ks /K), Ks∗ ) = 0 for any i ≥ 1. Proof.

This follows from 4.5.5 (ii), 4.5.6 and 4.5.7.

Theorem 4.5.10. Let L/K be an extension of fields, and let Ks and Ls be their separable closures, respectively. For any prime number p, we have cdp (Gal(Ls /L)) ≤ cdp (Gal(Ks /K)) + tr.deg(L/K), where tr.deg(L/K) denotes the transcendental degree of L over K. Proof. It suffices to treat the case where tr.deg(L/K) < ∞. There exists a subextension L0 /K of L/K such that L0 ∼ = K(t1 , . . . , tn ) is purely tran0 scendental, and L/L is algebraic. Note that Gal(Ls /L) can be regarded as a closed subgroup of Gal(L0s /L0 ), where L0s is a separable closure of L0 . By 4.4.3, we have cdp (Gal(Ls /L)) ≤ cdp (Gal(L0s /L0 )). It suffices to prove cdp (Gal(L0s /L0 )) ≤ cdp (Gal(Ks /K)) + tr.deg(L/K).

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So we may assume that L = K(t1 , . . . , tn ). By induction on n, we are reduced to the case where L = K(t). Let K(t)s be a separable closure of K(t). By 4.4.5, we have cdp Gal(K(t)s /K(t)) ≤ cdp Gal(K(t)s /Ks (t)) +cdp Gal(Ks (t)/K(t)) . Let K be an algebraic closure of K and let K(t)s be a separable closure of K(t). We have Gal(K(t)s /Ks (t)) ∼ = Gal(K(t)s /K(t)). By 4.5.8 and 4.5.9, we have cdp Gal(K(t)s /K(t)) ≤ 1. On the other hand, we have

Gal(Ks (t)/K(t)) ∼ = Gal(Ks /K). It follows that

cdp Gal(K(t)s /K(t)) ≤ 1 + cdp (Gal(Ks /K)).

This proves our assertion.

Theorem 4.5.11. Let k be a separably closed field and let K/k be an extension of transcendental degree 1. (i) We have H i (Gal(Ks /K), A) = 0 for any i ≥ 2 and any torsion Gal(Ks /K)-module A. (ii) We have H i (Gal(Ks /K), Ks∗ ) = 0 for i = 1 and any i ≥ 3. Let p be the characteristic of K. Then H 2 (Gal(Ks /K), Ks∗ ) is a p-torsion group. Proof. (i) By 4.5.10, for any prime number p, we have cdp (Gal(Ks /K)) ≤ cdp (Gal(ks /k)) + 1 = 1.

(ii) By 4.5.4, we have H 1 (Gal(Ks /K), Ks∗) = 0. By 4.4.2, we have i

scd(Gal(Ks /K)) ≤ cd(Gal(Ks /K)) + 1 ≤ 2.

So H (Gal(Ks /K), Ks∗ ) = 0 for any i ≥ 3. For any prime number ` distinct from p, we have an exact sequence of Gal(Ks /K)-modules x7→x`

0 → µ` (Ks ) → Ks∗ → Ks∗ → 0.

By (i), we have H 2 (Gal(Ks /K), µ`(Ks )) = 0. So `

H 2 (Gal(Ks /K), Ks∗ ) → H 2 (Gal(Ks /K), Ks∗ )

is injective. Hence H 2 (Gal(Ks /K), Ks∗ ) has no `-torsion, and must be a p-torsion group.

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Chapter 5

Etale Cohomology

5.1

ˇ Presheaves and Cech Cohomology

Let C be a category. A presheaf of sets (resp. abelian groups, rings, etc.) is a contravariant functor from C to the category of sets (resp. abelian groups, rings, etc.). Morphisms of presheaves are defined to be morphisms of functors. Given a presheaf F on C , an object U in C and a morphism f : U → V in C , elements in F (U ) are called sections of F on U , and the map F (f ) : F (V ) → F (U ) is called the restriction induced by f . For any s ∈ F (V ), we often denote F (f )(s) by s|U . In the following, we mainly discuss presheaves of abelian groups, and leave it for the reader to treat presheaves with other structures. We simply call a presheaf of abelian groups a presheaf. Denote the category of presheaves on C by PC . It is an abelian category. A sequence F →G →H of presheaves is exact if and only if for any object U in C , the sequence F (U ) → G (U ) → H (U ) is exact. Let f : C → C 0 be a covariant functor. It induces a covariant functor f P : PC 0 → PC ,

f P (F 0 )(U ) = F 0 (f (U )),

where F 0 is any object in PC 0 and U is any object in C . For any object U 0 in C 0 , define a category IU 0 as follows: Objects in IU 0 are pairs (U, φ) such that U are objects in C and φ : U 0 → f (U ) are morphisms in C 0 . Given two objects (Ui , φi ) (i = 1, 2) in IU 0 , a morphism ξ : (U1 , φ1 ) → (U2 , φ2 ) in IU 0 is a morphism ξ : U1 → U2 in C such that f (ξ)φ1 = φ2 . Let IU◦ 0 171

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be the opposite category of IU 0 . For any presheaf F on C , the assignment (U, φ) 7→ F (U ) defines a contravariant functor on IU 0 and hence a covariant functor on IU◦ 0 . Suppose for any presheaf F on C , and any object U 0 of C 0 , that lim(U,φ)∈ob I ◦ F (U ) exists. We define −→ U0 (fP F )(U 0 ) =

lim −→

F (U ).

◦ (U,φ)∈ob IU 0

Then fP F is a presheaf on C 0 . One can show that fP is left adjoint to the functor f P . That is, for any presheaf F on C and any presheaf F 0 on C 0 , we have a one-to-one correspondence Hom(fP F , F 0 ) ∼ = Hom(F 0 , f P F ) functorial in F and F 0 . Suppose fiber products exist in C . Let U be an object in C , and let U = {Uα → U }α∈I be a set of morphisms in C with a common target U . For any α0 , . . . , αn , let Uα0 ...αn = Uα0 ×U · · · ×U Uαn . For any presheaf F on C and any nonnegative integer n, let Y F (Uα0 ...αn ). C n (U, F ) = α0 ,...,αn

Define a homomorphism

d : C n (U, F ) → C n+1 (U, F ) as follows: For any s = (sα0 ...αn ) ∈ C n (U, F ), define ds = ((ds)α0 ...αn+1 ) ∈ C n+1 (U, F ) by (ds)α0 ...αn+1 =

n+1 X i=1

(−1)i sα0 ...αˆ i ...αn+1 |Uα0 ...αn+1 ,

where sα0 ...αˆ i ...αn+1 |Uα0 ...αn+1 is the image of sα0 ...αˆ i ...αn+1 under the restriction F (Uα0 ...αˆ i ...αn+1 ) → F (Uα0 ...αn+1 ) induced by the projection bαi ×U · · · ×U Uαn+1 . Uα0 ×U · · · ×U Uαn+1 → Uα0 ×U · · · ×U U

ˇ One can check that dd = 0. The complex (C · (U, F ), d) is called the Cech ˇ complex, and its cohomology groups are called the Cech cohomology groups ˇ · (U, F ). of F with respect to the family U, and are denoted by H

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ˇ Unlike the Cech cohomology for ordinary topological spaces, we cannot ˇ use alternative cochains to calculate the Cech cohomology for presheaves on categories. For example, if we take C to be the category of open subsets of a topological space X, U = {Uα → U }α∈I an open covering, U0 = ` { α∈I Uα → U }, and F a sheaf on X, then C · (U, F ) ∼ = C · (U0 , F ).

Since U0 has only one element, alternative n-cochains for U0 and F are ˇ n (U, F ) is not trivial. trivial for any n ≥ 1, whereas in general, H n It is clear that F → C (U, F ) is an exact functor on PC . Using this ˇ · (U, −) is a δ-functor. The following proposition fact, one checks that H shows that it is effaceable under mild conditions. Proposition 5.1.1. Assume that there exists a set Λ of objects in C such that for any presheaf A and any proper sub-presheaf B of A , there exists X ∈ Λ such that A (X) 6= B(X). Then the category PC has enough ˇ n (U, I ) = 0 for any injective presheaf I and any injective objects, and H n ≥ 1. Proof. For any object X in C , consider the category {X} with only one object X and only one morphism idX . Let i : {X} → C be the inclusion functor. For any abelian group M , denote the presheaf X 7→ M on {X} by M , and denote the presheaf iP M on C by MX . For any object U in C , we have M M. MX (U ) = HomC (U,X)

For any presheaf F on C , we have Hom(ZX , F ) ∼ = Hom(Z, iP F ) ∼ = Hom(Z, F (X)) ∼ = F (X). Taking into account of our assumption, we see that for any presheaf A and any proper sub-presheaf B of A , there exists X ∈ Λ such that there exists a morphism ZX → A which does not factor through B. So {ZX |X ∈ Λ} is a set of generators for PC . By the same argument as in the proof [Fu (2006)] 2.1.6 or [Grothendieck (1957)] 1.10.1, one can show that PC has enough injective objects. ˇ n (U , I ) = 0 for any injective presheaf I and any n ≥ 1. Let us prove H We need to show that the sequence Y Y Y d d d I (Uα0 ) → I (Uα0 α1 ) → I (Uα0 α1 α2 ) → · · · α0 ∈I

α0 ,α1 ∈I

α0 ,α1 ,α2 ∈I

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is exact. This sequence can be identified with Hom(

L

α0 ∈I

ZUα0 ,I )→Hom(

L

α0 ,α1 ∈I

ZUα0 α1 ,I )→Hom(

L

α0 ,α1 ,α2 ∈I

ZUα0 α1 α2 ,I )→··· .

Since I is an injective presheaf, it suffices to show that the sequence M M M δ1 δ2 δ0 ZUα0 α1 α2 ← ··· ZUα0 ← ZUα0 α1 ← α0 ,α1 ∈I

α0 ∈I

α0 ,α1 ,α2 ∈I

is an exact sequence of presheaves. We need to show that for any object X in C , the sequence M M M δ0 δ1 δ2 ZUα0 (X) ← ZUα0 α1 (X) ← ZUα0 α1 α2 (X) ← ··· α0 ∈I

α0 ,α1 ∈I

α0 ,α1 ,α2 ∈I

is exact, that is, the sequence L

L

δ

0 Z←

α0 ∈I Hom(X,Uα ) 0

L

L

δ

1 Z←

α0 ,α1 ∈I Hom(X,Uα α ) 0 1

L

L

δ

2 Z←···

α0 ,α1 ,α2 ∈I Hom(X,Uα α α ) 0 1 2

is exact. Let us describe the homomorphisms δn . For any f ∈ L L Hom(X, Uα0 ...αn ), let ef be the element in α0 ,...,αn ∈I Hom(X,Uα ...αn ) Z 0 whose component corresponding to f is 1 and whose other components are 0. Let bαi ×U · · · ×U Uαn+1 pi : Uα0 ×U · · · ×U Uαn+1 → Uα0 ×U · · · ×U U

be the projections. Then M δn :

M

α0 ,...,αn+1 ∈I Hom(X,Uα0 ...αn+1 )

Z→

M

M

Z

α0 ,...,αn ∈I Hom(X,Uα0 ...αn )

is defined by δn (ef ) =

n+1 X

(−1)i epi f .

i=0

Let πα (α ∈ I) be the morphisms Uα → U . For any φ ∈ Hom(X, U ), let a S(φ) = {g ∈ Hom(X, Uα )|πα g = φ}. α

The the above sequence can be written as L

L

φ∈Hom(X,U ) S(φ)

δ

0 Z←

L

L

δ

1 Z←

φ∈Hom(X,U ) S(φ)×S(φ)

L

L

δ

2 Z←··· .

φ∈Hom(X,U ) S(φ)×S(φ)×S(φ)

To prove that this sequence is exact, it suffices to show that for any φ, the sequence M δ0 M M δ1 δ2 Z← Z← Z← ··· S(φ)

S(φ)×S(φ)

S(φ)×S(φ)×S(φ)

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is exact. Here δn can be described as follows: For any (f0 , . . . , fn+1 ) ∈ L S(φ)n+2 , let e(f0 ,...,fn+1 ) be the element in S(φ)n+2 Z whose component corresponding to (f0 , . . . , fn+1 ) is 1 and whose other components are 0. Then M M δn : Z→ Z S(φ)n+2

S(φ)n+1

is defined by

δn (e(f0 ,...,fn+1 ) ) =

n+1 X

(−1)i e(f0 ,...,fˆi ,...,fn+1 ) .

i=0

Let

δ−1 :

M

S(φ)

Z→Z

be the homomorphism defined by δ0 (ef ) = 1 for any f ∈ S(φ). Fix an element g in S(φ). Consider the homomorphisms M M Dn : Z→ Z, Dn (e(f0 ,...,fn+1 ) ) = e(g,f0 ,...,fn+1 ) (n ≥ 0), S(φ)n+1

S(φ)n+2

D−1 : Z →

M

D−1 (1) = eg .

Z,

S(φ)

Then Dn define a homotopy between the identity and the zero morphism of the complex M M δ−1 M δ2 δ1 δ0 Z← ··· . Z← Z← 0←Z ← S(φ)

S(φ)×S(φ)

S(φ)×S(φ)×S(φ)

Hence this complex is acyclic.

Let U = {Uα → U }α∈I and U0 = {Uα0 0 → U }α0 ∈I 0 be two sets of morphisms in C with a common target U . A morphism f : U → U0 consists 0 of a map : I → I 0 and a morphism fα : Uα → U(α) in C for each α ∈ I. Such a morphism induces a morphism of complexes f ∗ : C · (U0 , F ) → C · (U, F )

as follows: Given s = (sα00 ...α0n ) ∈ C n (U0 , F ), define f ∗ s = ((f ∗ s)α0 ...αn ) ∈ C n (U, F ) by (f ∗ s)α0 ...αn = s(α0 )...(αn ) |Uα0 ...αn ,

where s(α0 )...(αn ) |Uα0 ...αn is the image of s(α0 )...(αn ) under the restriction induced by the morphism 0 0 fα0 × · · · × fαn : Uα0 ×U · · · ×U Uαn → U(α ×U · · · ×U U(α . 0) n)

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Let g : U → U0 be another morphism consisting of a map δ : I → I 0 and 0 morphisms gα : Uα → Uδ(α) (α ∈ I). Define homomorphisms H : C n (U0 , F ) → C n−1 (U, F )

as follows: Given s = (sα00 ...α0n ) ∈ C n (U0 , F ), define Hs = ((Hs)α0 ...αn−1 ) ∈ C n−1 (U, F ) by n X (Hs)α0 ...αn−1 = (−1)i s(α0 )...(αi )δ(αi )...δ(αn−1 ) |Uα0 ...αn−1 , i=0

where s(α0)...(αi)δ(αi)...δ(αn−1) |Uα0 ...αn−1 is the image of s(α0)...(αi)δ(αi)...δ(αn−1) under the restriction induced by the morphism fα0 × · · · × fαi−1 × (fαi , gαi ) × gαi+1 × · · · × gαn−1 :

0 0 0 0 Uα0 ×U · · · ×U Uαn−1 → U(α ×U · · · ×U U(α ×U Uδ(α ×U · · · ×U Uδ(α . 0) i) i) n−1)

Then we have Hd + dH = g ∗ − f ∗ .

So f ∗ and g ∗ are homotopic to each other, and hence induce the same ˇ · (U0 , F ) → H ˇ · (U, F ) on Cech ˇ homomorphisms H cohomology groups. We say that U is a refinement of U0 if there exists a morphism from U to U0 . ˇ · (U0 , F ) → H ˇ · (U, F ) on Cech ˇ So we have canonical homomorphisms H 0 cohomology groups whenever U is a refinement of U . 5.2

Etale Sheaves

([SGA 4] VII 1–2, [SGA 4 21 ] Arcata I, II.) Let X be a scheme. Define a category Xet as follows: Objects in Xet are X-schemes U so that the structure morphisms U → X are etale. For any two etale X-schemes U1 and U2 , morphisms from U1 to U2 in Xet are Xmorphisms from U1 to U2 . Note that morphisms in Xet are etale, and fiber products exist in Xet . An etale presheaf on X is a presheaf on the category Xet . The category of etale presheaves on X is denoted by PX . It is an abelian category. Let f : X 0 → X be a morphism of schemes. It induces a functor 0 fet : Xet → Xet ,

U 7→ U ×X X 0 .

For any etale presheaf F 0 on X 0 , denote the etale presheaf (fet )P (F 0 ) by fP F 0 . We have (fP F 0 )(U ) = F 0 (U ×X X 0 )

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0 for any U ∈ ob Xet . For any object U 0 in Xet , define a category IU 0 as follows: Object in IU 0 are pairs (U, φ), where U is an object in Xet , and φ : U 0 → U is an X-morphism. For any two such pairs (Ui , φi ) (i = 1, 2), a morphism ξ : (U1 , φ1 ) → (U2 , φ2 ) in IU 0 is an X-morphism ξ : U1 → U2 such that ξφ1 = φ2 . Note that this category is equivalent to the category 0 defined in 5.1 for the functor fet : Xet → Xet . We will show shortly that ◦ the opposite category IU 0 of IU 0 satisfies the conditions (I2) and (I3) in 2.7, and that there exists a cofinal subcategory of IU◦ 0 whose objects form a set. For any etale presheaf F on X, denote the etale presheaf (fet )P (F ) on X 0 by f P F . We have

(f P F )(U 0 ) =

lim −→

F (U )

◦ (U,φ)∈obIU 0

0 for any U 0 ∈ ob Xet . The functor f P : PX → PX 0 is exact, and is left adjoint to the functor fP : PX 0 → PX , that is, for any etale presheaf F on X and any etale presheaf F 0 on X 0 , we have a canonical one-to-one correspondence

Hom(f P F , F 0 ) ∼ = Hom(F , fP F 0 ) functorial in F and F 0 . Let us show that IU◦ 0 satisfies the conditions (I2) and (I3). Given two morphisms α, β : (U2 , φ2 ) → (U1 , φ1 ) in IU 0 , consider the Cartesian diagram p2

K = U2 ×(U2 ×X U1 ) U2 → p1 ↓ Γ

U2 ↓ Γβ

α U2 → U2 ×X U1 ,

where Γα and Γβ are the graphs of α and β, respectively. Note that the schemes in this diagram are etale X-schemes. Let π1 : U2 ×X U1 → U2 be the projection. We have p1 = π1 Γα p1 = π1 Γβ p2 = p2 . Denote p1 = p2 by p. One can show that the sequence p

α

K → U2 ⇒ U1 β

is exact, that is, αp = βp, and that for any morphism ψ : V → U2 of schemes with the property αψ = βψ, there exists a unique morphism ψ 0 : V → K such that pψ 0 = ψ. We have αφ2 = βφ2 = φ1 .

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So there exists a morphism φ : U 0 → K such that pφ = φ2 . The morphism p defines a morphism p : (K, φ) → (U2 , φ2 ) in IU 0 and we have αp = βp. So IU◦ 0 satisfies (I2). Given two objects (Ui , φi ) (i = 1, 2) in IU 0 , let pi : U1 ×X U2 → Ui be the projections, and let (φ1 , φ2 ) : U 0 → U1 ×X U2 be the morphism with the property pi ◦ (φ1 , φ2 ) = φi . Then the pair (U1 ×X U2 , (φ1 , φ2 )) is an object in IU 0 , and pi define morphisms pi : (U1 ×X U2 , (φ1 , φ2 )) → (Ui , φi ) in IU 0 . So IU◦ 0 satisfies (I3). Next we show IU◦ 0 has a cofinal subcategory whose objects form a set. For any u0 ∈ U 0 , choose an affine open subset Xu0 of X containing the image of u0 in X. We will construct a set S of X-schemes Y with the following property (*) so that any X-scheme with the property (*) is isomorphic to an X-scheme in S : (*) Y can be covered by affine open subsets Yu0 (u0 ∈ U 0 ) so that for each u0 , the image of Yu0 in X is contained in Xu0 and that Γ(Yu0 , OY ) is a Γ(Xu0 , OX )-algebra of finite presentation. Then objects (U, φ) in IU◦ 0 with U ∈ S (and φ ∈ HomX (U 0 , U )) form a set, and the full subcategory of IU◦ 0 consisting of these objects is cofinal in IU◦ 0 . Let Su0 be the family of affine schemes Spec (Γ(Xu0 , OX )[t1 , . . . , tn ]/I) with n ∈ N and with I being finitely generated ideals of Γ(Xu0 , OX )[t1 , . . . , tn ]. Then Su0 is a set. So Tu0 = {(Z, W )|Z ∈ Su0 , W is an open subset of Z} is a set. Hence { (Zu0 , Wu0 v0 , ψu0 v0 )u0 ,v0 ∈U 0 | (Zu0 , Wu0 v0 ) ∈ Tu0 ,

ψu0 v0 ∈ HomX (Wu0 v0 , Wv0 u0 ), Wu0 u0 = Zu0 , ψu0 u0 = id,

ψu0 v0 (Wu0 w0 ∩ Wu0 v0 ) ⊂ Wv0 w0 ∩ Wv0 u0 ,

ψv0 w0 ψu0 v0 = ψu0 w0 on Wu0 v0 ∩ Wu0 w0 }

is a set. This set is nothing but the set of gluing data for X-schemes with the property (*). We take S to be the set of X-schemes defined by gluing data in this set. Let X be a scheme. A set U = {Uα → U }α∈I of morphisms in Xet with a common target U is called an etale covering of U if U is the union of

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images of Uα . An etale presheaf F on X is called an etale sheaf if for any etale covering U = {Uα → U }α∈I , the sequence Y Y 0 → F (U ) → F (Uα ) → F (Uα ×U Uβ ) α∈I

α,β∈I

is exact, where the homomorphisms in this sequence are defined by Y F (U ) → F (Uα ), s 7→ (s|Uα ), Y

α∈I

F (Uα ) →

Y

α,β∈I

α∈I

F (Uα ×U Uβ ),

(sα ) 7→ (sβ |Uα ×U Uβ − sα |Uα ×U Uβ ).

Morphisms between etale sheaves are defined to be morphisms between them that are considered as presheaves. Denote the category of etale sheaves on X by SX . It is a full subcategory of the category PX of etale presheaves on X. Denote by i : SX → PX be the inclusion. ¿From now on, we call etale sheaves and etale presheaves simply sheaves and presheaves. Proposition 5.2.1. There exists a functor # : PX → SX that is left adjoint to i : SX → PX , that is, for any presheaf G on X, there exists a pair (G # , θ) consisting of a sheaf G # on X and a morphism θ : G → G # of presheaves such that for any sheaf F on X and any morphism of presheaves φ : G → F , there exists one and only one morphism ψ : G # → F such that ψθ = φ. The pair (G # , θ) is unique up to a unique isomorphism. We call G # the sheaf associated to the presheaf G . To prove the proposition, we first define a functor + : PX → PX . For every object U in Xet , let JU be the category defined as follows: Objects in JU are etale coverings of U . Morphisms in JU are defined in the same way as in the end of 5.1. We say that an etale covering U is a refinement of an etale covering U0 and write U0 ≤ U if there exists a morphism U → U0 in JU . One can show that (JU , ≤) is a directed family, and that it has a cofinal subfamily whose objects form a set. By the discussion at the end of 5.1, we have a well-defined homomorphism ˇ 0 (U0 , G ) → H ˇ 0 (U, G ) H for any presheaf G on X. We define G + (U ) =

lim −→

U∈ob JU

ˇ 0 (U, G ). H

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Let V → U be a morphism in Xet , and let U = {Uα → U }α∈I be an etale covering of U . Then V = {Uα ×U V → V }α∈I is an etale covering of V . ˇ We have a canonical morphism of Cech complexes C · (U, G ) → C · (V, G ). It induces a homomorphism ˇ 0 (U, G ) → H ˇ 0 (V, G ) H and hence a homomorphism ˇ 0 (U, G ) → G + (V ) = G + (U ) = lim H −→ U∈ob JU

We thus get a presheaf G

+

lim −→

ˇ 0 (V, G ). H

V∈ob JV

on X. The functor

+ : PX → PX ,

G 7→ G +

is leaf exact. Note that U0 = {id : U → U } is an etale covering, and that any etale covering U of U is a refinement of U0 . We have ˇ 0 (U0 , G ) = G (U ). H So we have a canonical homomorphism ˇ 0 (U, G ) = G + (U ). G (U ) → lim H −→ U∈ob JU

We thus have a morphism of presheaves G → G + . If G is a sheaf, then this morphism is an isomorphism. Lemma 5.2.2. For any presheaf G and any sheaf F on X, the canonical morphism G → G + induces a one-to-one correspondence Φ : Hom(G + , F ) ∼ = Hom(G , F ). Proof.

Any homomorphism φ : G → F induces a morphism φ+ : G + → F + ∼ = F.

We thus have a map Ψ : Hom(G , F ) → Hom(G + , F ),

φ 7→ φ+ .

It is clear that ΦΨ = id. Let U = {Uα → U }α∈I be an etale covering. For any α0 ∈ I and any morphism ψ : G + → F , we have a commutative diagram ψ

G (U ) → G + (U ) → F (U ) ↓ ↓ ↓ ψ

G (Uα0 ) → G + (Uα0 ) → F (Uα0 ).

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Note that the etale covering V0 = {id : Uα0 → Uα0 } is a refinement of the etale covering V = {Uα ×U Uα0 → Uα0 }α∈I . Indeed, the diagonal morphism ∆ : Uα0 → Uα0 ×U Uα0 defines a morphism V0 → V. Given Y Y ˇ 0 (U, G ) = ker ξ = (ξα ) ∈ H G (Uα ) → G (Uα ×U Uβ ) , α

α,β

one can check the image of ξ under the composite ˇ 0 (U, G ) → H ˇ 0 (V, G ) → H ˇ 0 (V0 , G ) = G (Uα0 ) H

ˇ 0 (U, G ) in G + (U ) = lim ˇ 0 (U, G ). is ξα0 . Let ξ¯ be the image of ξ ∈ H H −→U∈ob JU Then the image of ξ¯ under the homomorphism G + (U ) → G + (Uα ) is the 0

same as the image of ξα0 under the homomorphism G (Uα0 ) → G + (Uα0 ). ¯ Uα ∈ F (Uα0 ) coincides with the image of ξα0 under the composite So ψ(ξ)| 0 G (Uα0 ) → G + (Uα0 ) → F (Uα0 ).

This is true for any α0 ∈ I. From this fact, one deduces ΨΦ(ψ) = ψ.

Lemma 5.2.3. We say that a presheaf G on X has the property (+) if for any etale covering {Uα → U }α∈I in Xet , the canonical homomorphism Y G (U ) → G (Uα ) α∈I

is injective. (i) For any presheaf G , G + has the property (+). (ii) If a presheaf G has the property (+), then G + is a sheaf.

Proof. (i) Let {Uα → U }α∈I be an etale covering, and let ξ¯i ∈ G + (U ) (i = 1, 2) such that they have the same image under the homomorphism Y G + (U ) → G + (Uα ). α∈I

Choose an etale covering V = {Vβ → U }β∈J such that ξ¯i are the images of Y Y ˇ 0 (V, G ) = ker ξi ∈ H G (Vβ ) → G (Vβ1 ×U Vβ2 ) β∈J

under the homomorphism ˇ 0 (V, G ) → H

β1 ,β2 ∈J

ˇ 0 (U, G ) = G + (U ). lim H −→

U∈ob JU

For each i ∈ {1, 2} and each α ∈ I, the image of ξ¯i under the homomorphism G + (U ) → G + (Uα ) is the image of ξi under the composite ˇ 0 (V, G ) → H ˇ 0 (V ×U Uα , G ) → G + (Uα ), H

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where V ×U Uα = {Vβ ×U Uα → Uα }β∈J . Since ξ¯i (i = 1, 2) have the same image in G + (Uα ), there exists an etale covering Wα = {Wαγ → Uα }γ∈Kα which is a refinement of V ×U Uα such that the images of ξi (i = 1, 2) under the composite ˇ 0 (V, G ) → H ˇ 0 (V ×U Uα , G ) → H ˇ 0 (Wα , G ) H are the same, that is, ξi (i = 1, 2) have the same image in Y Y ˇ 0 (Wα , G ) = ker H G (Wαγ ) → G (Wαγ1 ×Uα Wαγ2 ) . γ∈Kα

γ1 ,γ2 ∈Kα

But {Wαγ → Uα → U }α∈I,γ∈Kα is an etale covering of U , which is a refinement of V. Denote this etale covering by W. Then ξi (i = 1, 2) have the same image under the map ˇ 0 (V, G ) → H ˇ 0 (W, G ) H Y = ker

α∈I,γ∈Kα

G (Wαγ ) →

Y

α1 , α2 ∈ I, γ1 ∈ K α1 , γ2 ∈ K α2

G (Wα1 γ1 ×U Wα2 γ2 ) .

So ξi (i = 1, 2) have the same image in G + (U ), that is, ξ¯1 = ξ¯2 . This proves Q that the homomorphism G + (U ) → α∈I G + (Uα ) is injective. (ii) We claim that if G has the property (+), then for any etale covering U = {Uα → U }α∈I of U ∈ ob Xet and any refinement V = {Vβ → U }β∈J ˇ 0 (U, G ) → H ˇ 0 (V, G ) is injective. of U, the canonical homomorphism H Let f : V → U be a morphism. Then W = {Uα ×U Vβ → U }α∈I,β∈J is an etale covering U and the projections Uα ×U Vβ → Uα and Uα ×U Vβ → Vβ define morphisms p1 : W → U and p2 : W → V. For each fixed α ∈ I, {Uα ×U Vβ → Uα }β∈J is an etale covering of Uα . Since G has the property (+), the canonical homomorphism Y G (Uα ) → G (Uα ×U Vβ ) β∈J

is injective. So

Y

α∈I

G (Uα ) →

Y

α∈I, β∈J

G (Uα ×U Vβ )

is injective, and hence ˇ 0 (U, G ) → H ˇ 0 (W, G ) p∗1 : H

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is injective. This last homomorphism coincides with the composite ∗

∗

p2 f ˇ 0 (U, G ) → ˇ 0 (V, G ) → ˇ 0 (W, G ). H H H ˇ 0 (U, G ) → H ˇ 0 (V, G ) is injective. This prove our claim. It follows that H + To prove that G is a sheaf, we need to show that for any etale covering {Uα → U }α∈I , the sequence Y Y 0 → G + (U ) → G + (Uα ) → G + (Uα ×U Uβ ) α∈I

is exact. Since G

α,β∈I

+

has the property (+), we only need to show that any Y Y ξ¯ = (ξ¯α ) ∈ ker G + (Uα ) → G + (Uα ×U Uβ ) α∈I

α,β∈I

Q lies in the image of the homomorphism G (U ) → α∈I G + (Uα ). For each α ∈ I, choose an etale covering {Uαλ → Uα }λ∈Iα such that ξ¯α is the image in G + (Uα ) of ˇ 0 ({Uαλ → Uα }, G ) ξα = (ξαλ ) ∈ H Y Y = ker G (Uαλ ) → G (Uαλ1 ×Uα Uαλ2 ) . +

λ∈Iα

λ1 ,λ2 ∈Iα

For each pair α, β ∈ I, by the base change Uα ×U Uβ → Uα , ξα induces an element

∈

1 1 ξαβ = (ξαβλ )

ker

Q

λ∈Iα

Q G (Uαλ ×U Uβ )→ λ

1 ,λ2 ∈Iα

and by the base change Uα ×U Uβ → Uβ , ξβ induces an element 2 2 ξαβ = (ξαβµ )

∈ ker

Q

µ∈Iβ

Q G (Uα ×U Uβµ )→ µ

1 ,µ2 ∈Iβ

G ((Uαλ1 ×U Uβ )×(Uα ×U Uβ ) (Uαλ2 ×U Uβ )) ,

G ((Uα ×U Uβµ1 )×(Uα ×U Uβ ) (Uα ×U Uβµ2 )) ,

1 2 where ξαβλ are the restrictions of ξαλ ∈ G (Uαλ ) to Uαλ ×U Uβ , and ξαβµ are the restrictions of ξβµ ∈ G (Uβµ ) to Uα ×U Uβµ .

{Uαλ ×U Uβµ }λ∈Iα ,µ∈Iβ → {Uαλ ×U Uβ }λ∈Iα → {Uαλ }λ∈Iα ↓ ↓ ↓ {Uα ×U Uβµ }µ∈Iβ → Uα ×U Uβ → Uα ↓ ↓ ↓ {Uβµ }µ∈Iβ → Uβ → U

Since we have Y Y ξ¯ = (ξ¯α ) ∈ ker G + (Uα ) → G + (Uα ×U Uβ ) , α∈I

α,β∈I

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1 2 the images of ξαβ and ξαβ in G + (Uα ×U Uβ ) are the same. So there exists an etale covering V of Uα ×U Uβ which is a common refinement of the etale coverings {Uαλ ×U Uβ → Uα ×U Uβ }λ∈Iα and {Uα ×U Uβµ → Uα ×U Uβ }µ∈Iβ 1 2 ˇ 0 (V, G ). By the claim at such that ξαβ and ξαβ have the same image in H 1 2 ˇ 0 (V, G ) the beginning of the proof, ξαβ and ξαβ have the same image in H for any common refinement V of the above two etale coverings. Taking

V = {Uαλ ×U Uβµ → Uα ×U Uβ }λ∈Iα , µ∈Iβ , 1 2 we see that ξαβλ and ξαβµ have the same image in G (Uαλ ×U Uβµ ). So ξαλ ∈ G (Uαλ ) and ξβµ ∈ G (Uβµ ) have the same image in G (Uαλ ×U Uβµ ), and hence we have Y Y (ξαλ ) ∈ ker G (Uαλ ) → G (Uαλ ×U Uβµ ) α∈I,λ∈Iα

α, β ∈ I, λ ∈ Iα , µ ∈ Iβ

ˇ 0 ({Uαλ → U }α∈I,λ∈Iα , G ). =H One then checks that the image of (ξαλ ) under the homomorphism ˇ 0 ({Uαλ → U }α∈I,λ∈Iα , G ) → G + (U ) H Q is mapped to ξ¯ = (ξ¯α ) under G + (U ) → α∈I G + (Uα ).

Proof of 5.2.1. For any presheaf G on X, the presheaf G ++ is a sheaf by 5.2.3. We have a canonical morphism G → G ++ . By 5.2.2, it induces a one-to-one correspondence Hom(G ++ , F ) ∼ = Hom(G , F ) for any sheaf F on X. It suffices to take G # = G ++ .

Proposition 5.2.4. (i) SX is an abelian category with enough injective objects. Let φ : F → G be a morphism of sheaves on X. Then ker φ is the sheaf defined by U 7→ ker(φ(U ) : F (U ) → G (U )) for any U ∈ ob Xet , and coker φ (resp. im φ, resp. coim φ) is the sheaf associated to the presheaf defined by U 7→ coker(φ(U ))

(resp. U 7→ im(φ(U )),

resp. U 7→ coim(φ(U ))).

(ii) The inclusion i : SX → PX is left exact, and # : PX → SX is exact.

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Proof. (i) Let φ : F → G be a morphism of sheaves. The kernel of φ in the category of presheaves is a sheaf, and it is the kernel of φ in the category of sheaves. Let C be the cokernel of φ in the category of presheaves. One can verify that C # is the cokernel of φ in the category of sheaves. Let P be the cokernel of the canonical morphism ker φ → F in the category of presheaves. We have an exact sequence 0→P →G →C →0 in the category of presheaves. Since the functor + : PX → PX is left exact, and # = + ◦ +, the sequence 0 → P# → G # → C # is exact in the category of presheaves. It follows that P# ∼ = ker (G → C # ). Note that P # is the coimage of φ and ker (G → C # ) is the image of φ in the category of sheaves. So SX is an abelian category. For every object U in Xet , we have the presheaf ZU on X defined in the proof of 5.1.1. Denote the sheaf (ZU )# also by ZU . We can find a set of objects in Xet such that for any sheaf A and any proper subsheaf B of A , there exists an object U in this set such that A (U ) 6= B(U ). Then {ZU } is a set of generators for SX . By the same argument as in the proof of [Fu (2006)] 2.1.6 or [Grothendieck (1957)] 1.10.1, one can show that SX has enough injective objects. (ii) We have seen that i and # is left exact in the proof of (i). Given an exact sequence 0 → G 0 → G → G 00 → 0 in the category of presheaves, for any sheaf F , the sequence 0 → Hom(G 00 , F ) → Hom(G , F ) → Hom(G , F ) is exact. Hence the sequence 0 → Hom(G 00# , F ) → Hom(G # , F ) → Hom(G # , F ) is exact. It follows that G 0# → G # → G 00# → 0 is exact in the category of sheaves. So # is exact.

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Let f : X 0 → X be a morphism of schemes. For any sheaf F 0 on X 0 , the presheaf fP F 0 is a sheaf. We denote it by f∗ F 0 and call it the direct image of F 0 . For any object U in Xet , we have f∗ (F 0 ) = F 0 (U ×X X 0 ). For any sheaf F on X, define the inverse image f ∗ F of F on X 0 to be the sheaf (f P F )# . We often denote f ∗ F by F |X 0 . The functor f ∗ is left adjoint to f∗ , that is, we have a one-to-one correspondence Hom(f ∗ F , F 0 ) ∼ = Hom(F , f∗ F 0 ) functorial in F and F 0 . Proposition 5.2.5. (i) For any presheaf G on X, we have f ∗ (G # ) ∼ = (f P G )# . ∗ (ii) f∗ : SX 0 → SX is left exact and f : SX → SX 0 is exact. (iii) Suppose f : X 0 → X is etale. Then for any sheaf F on X, f ∗ F is the sheaf defined by (f ∗ F )(U 0 ) = F (U 0 ) 0 for any object U 0 in Xet , where on the right-hand side, we regard U 0 as an etale X-scheme. (iv) Let f : X 0 → X and g : X 00 → X 0 be two morphisms of schemes. We have (f g)∗ ∼ = g∗f ∗. = f∗ g∗ and (f g)∗ ∼

Proof. (i) For any sheaf F 0 on X 0 , we have one-to-one correspondences Hom(f ∗ (G # ), F 0 ) ∼ = Hom(G # , f∗ F 0 ) ∼ = Hom(G , fP F 0 ) ∼ = Hom(f P G , F 0 )

∼ = Hom((f P G )# , F 0 ).

So we have (f P G )# ∼ = f ∗ (G # ). (ii) The left exactness of f∗ follows from the definition. Let 0 → F1 → F2 → F3 → 0 be an exact sequence of sheaves on X. Let C be cokernel of F2 → F3 in the category of presheaves. Then the sequence 0 → F1 → F2 → F3 → C → 0

etale

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is exact in the category of presheaves and C # = 0. Since f P is exact, the sequence 0 → f P F1 → f P F2 → f P F3 → f P C → 0 is exact in the category of presheaves. Since # is exact, the sequence 0 → (f P F1 )# → (f P F2 )# → (f P F3 )# → (f P C )# → 0 is exact in the category of sheaves. By (i), we have (f P C )# ∼ = f ∗ (C # ) = 0. It follows that 0 → (f P F1 )# → (f P F2 )# → (f P F3 )# → 0 is exact in the category of sheaves. Hence f ∗ is exact. (iii) We have (f P F )(U 0 ) =

lim −→

F (U ).

◦ (U,φ)∈ob IU 0

One can show that (U 0 , id) is a final object in the category IU◦ 0 . So we have (f P F )(U 0 ) = F (U 0 ). One easily shows that f P F , given by this formula, is a sheaf. So we have (f ∗ F )(U 0 ) = F (U 0 ). (iv) We have (f g)∗ ∼ = g∗f ∗ = f∗ g∗ by definition. This implies that (f g)∗ ∼ by adjunction. Proposition 5.2.6. Let F be a presheaf on a scheme X. Suppose for any etale covering {Uα → U }α∈I with the property (a) or (b) below, that the sequence Y Y 0 → F (U ) → F (Uα ) → F (Uα ×U Uβ ) α∈I

α,β∈I

is exact: (a) Uα and U are affine and I is finite. (b) Uα → U are open immersions for all α. Then F is a sheaf.

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Proof. Let U = {Uα → U }α∈I be an etale covering of U , let {Vλ }λ∈J be a covering of U by affine open subsets, and let Uλ = {Uα ×U Vλ → Vλ }α∈I for each λ ∈ J. There exists a refinement U0λ of Uλ satisfying the condition (a). By our assumption, we have ˇ 0 (U0 , F ). F (Vλ ) ∼ =H λ

This isomorphism coincides with the composite ˇ 0 (Uλ , F ) → H ˇ 0 (U0λ , F ). F (Vλ ) → H

So

ˇ 0 (Uλ , F ) F (Vλ ) → H

is injective. We have a commutative diagram Q F (U ) → α∈I F (Uα ) ↓ ↓ Q Q Q F (V ) → F (Uα ×U Vλ ). λ λ∈I α∈I λ∈I The vertical arrows are injective by (b). We have just shown that the bottom horizontal arrow is injective. So Y F (U ) → F (Uα ) α∈I

is injective. Hence F has the property (+). By the claim at the beginning of the proof of 5.2.3 (ii), ˇ 0 (Uλ , F ) → H ˇ 0 (U0λ , F ) H

is injective. Since the composite ˇ 0 (Uλ , F ) → H ˇ 0 (U0λ , F ) F (Vλ ) → H is an isomorphism, we have

ˇ 0 (Uλ , F ). F (Vλ ) ∼ =H Consider the commutative diagram

0→

0

0

0

↓

↓

↓

F (U)

Q

→

F (Uα )

Q

→

α∈I

↓ Q

0→

F (Vλ )

↓ Q Q

→

↓ 0→

Q

λ,µ∈J

F (Uα ×U Vλ )

↓ Q

→

λ∈J α∈I

λ∈J

Q

Q

Q

λ,µ∈J α∈I

F (Uα ×U Uβ ×U Vλ )

λ∈J α,β∈I

↓

F (Vλ ∩Vµ )→

F (Uα ×U Uβ )

α,β∈I

F (Uα ×U (Vλ ∩Vµ ))→

↓ Q

Q

F (Uα ×U Uβ ×U (Vλ ∩Vµ )).

λ,µ∈J α,β∈I

By (b), the vertical sequences are exact. We have just shown that the second horizontal sequence is exact. By a diagram chasing, one can show the first horizontal sequence is exact. So F is a sheaf.

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Let X be a scheme and let Y be an X-scheme. Consider the presheaf Ye on X defined by Ye (U ) = HomX (U, Y )

for any object U in Xet . Let {Uα → U }α∈I be an etale covering. If Uα and ` U are affine and I is finite, then α∈I Uα → U is a quasi-compact faithfully flat morphism. By 1.8.3, the sequence Y Y HomX (U, Y ) → HomX (Uα , Y ) ⇒ HomX (Uα ×U Uβ , Y ) α∈I

α,β∈I

is exact. If each Uα → U is an open immersion, then the above sequence is also exact. By 5.2.6, Ye is a sheaf of sets on X. We call it the sheaf represented by the X-scheme Y . Proposition 5.2.7. Let Y be an etale X-scheme. (i) For any sheaf F of sets on X, we have a one-to-one correspondence Hom(Ye , F ) ∼ = F (Y ). e for some open subset U of Y . (ii) Any subsheaf of Ye is of the form U 0 (iii) Let f : X → X be a morphism. We have f ∗ Ye ∼ = (Y ×X X 0 )∼ .

Proof. (i) Given a morphism of sheaves φ : Ye → F , let S(φ) ∈ F (Y ) be the image of idY under the map φ(Y ) : Hom(Y, Y ) = Ye (Y ) → F (Y ).

Given a section s ∈ F (Y ), define a morphism T (s) : Ye → F so that for any object U in Xet and any t ∈ Ye (U ), the image of t under the map T (s)(U ) : Ye (U ) → F (U )

is the image of s under the restriction F (Y ) → F (U ) induced by the morphism t : U → Y . Then S : Hom(Ye , F ) → F (Y ) and T : F (Y ) → Hom(Ye , F ) are inverses to each other. (ii) Let F be a subsheaf of Ye . Objects in Yet can be considered as objects in Xet . Let {Uα }α∈I be the family of all objects in Yet such that the structure morphisms Uα → Y lie in the subset F (Uα ) of HomX (Uα , Y ). Let U be the union of the images of Uα in Y . Then for any object V in Xet , we have F (V ) ⊂ HomX (V, U ).

e . By our construction, {Uα → U }α∈I is a covering So F is a subsheaf of U and the image of the inclusion iU : U ,→ Y under the maps Ye (U ) → Y˜ (Uα )

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lie in F (Uα ) for all α ∈ I. So we have iU ∈ F (U ). It follows that e (U ) lies in F (U ). For any object V in Xet , if U e (V ) = Ø, then we idU ∈ U e . If U e (V ) 6= Ø, then an arbitrary have F (V ) = Ø since F is a subsheaf of U e e (U ) → U e (V ) section s ∈ U (V ) is the image of idU under the restriction U induced by s : V → U . As idU ∈ F (U ), we have s ∈ F (V ). We thus have e (V ), and hence F = U e. F (V ) = U A flaw in the above proof is that the family {Uα }α∈I may not be a set. To avoid this, one fixes an affine open covering {Vj }j∈J of Y with indices j lying in a set J. For each j ∈ J, one can find a set of objects {Ujβ }β∈Ij in Yet such that each Ujβ is affine, its image in Y is contained in Vj , and any affine object in (Vj )et is Y -isomorphic to some Ujβ . Then the above argument works if we take {Uα }α∈I to be the set consisting of those Ujβ (j ∈ J, β ∈ Ij ) such that the structure morphisms Ujβ → Y lie in F (Ujβ ). (iii) For any sheaf F 0 on X 0 , we have Hom(f ∗ Ye , F 0 ) ∼ = Hom(Ye , f∗ F 0 ) ∼ = (f∗ F 0 )(Y ) ∼ = F 0 (Y ×X X 0 ) ∼ = Hom((Y ×X X 0 )∼ , F 0 ).

So we have f ∗ Ye ∼ = (Y ×X X 0 )∼ .

Let X be a scheme and let M be a quasi-coherent OX -module. By 1.6.2 and 5.2.6, the presheaf on X defined by U 7→ Γ(U, π ∗ M )

for any object π : U → X in Xet is a sheaf. We denote the sheaf by Met . Taking M = OX , we get the sheaf OXet . We have OXet (U ) = Γ(U, OU ). The subsheaf of OXet defined by OXet (U ) = Γ(U, OU∗ ) ∗ . is denoted by OX et f Suppose that X is quasi-compact and quasi-separated. Let Xet be the full subcategory of Xet whose objects are etale X-schemes with finite pref f sentation. Note that fiber products exist in Xet . An etale covering in Xet is defined to be a family of morphisms {Uα → U }α∈I such that I is finite and f U is the union of the images of Uα . A presheaf F on Xet is called a sheaf if for any etale covering {Uα → U }α∈I as above, the canonical sequence Y Y F (Uα ) → F (Uα ×U Uβ ) 0 → F (U ) → α∈I

α,β∈I

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is exact. Denote the category of presheaves and the category of sheaves on f f Xet by PX f and SX f , respectively. Let i : Xet → Xet be the inclusion. It induces functors iP : PX → PX f ,

iP : PX f → PX .

f If F is a sheaf on X, then iP F is a sheaf on Xet . Let r : SX → SX f be P the restriction of i to the category of sheaves.

Proposition 5.2.8. Suppose that X is quasi-compact and quasi-separated. Then the functor r : SX → SX f defines an equivalence of categories. Proof.

Let F and G be two sheaves on X. Consider the map HomSX (F , G ) → HomSX f (F , G ).

Suppose that φi : F → G (i = 1, 2) are two morphisms which have the same image in HomSX f (F , G ). Then for any object U in Xet with finite presentation, φi (i = 1, 2) induce the same homomorphism F (U ) → G (V ). For a general object U in Xet , let {Uα }α∈I be a covering of U by affine open subsets. Then each Uα has finite presentation over X. Since Y Y F (U ) → F (Uα ), G (U ) → G (Uα ) α∈I

α∈I

are injective and φi (i = 1, 2) induce the same homomorphism F (Uα ) → G (Uα ) for each α, φi induce the same homomorphism F (U ) → G (U ). So we have φ1 = φ2 . This proves that the above map is injective. We leave it for the reader to prove the map is surjective. f Let F be a sheaf on Xet . We claim that ∼ F. r((iP F )# ) = This proves that the functor r is essentially surjective. Let U be an object f in Xet . It can also be regarded as an object in Xet . Using the definition of the functor iP , one can verify that (iP F )(U ) = F (U ). For any etale f covering of U in Xet , there exists an etale covering of U in Xet refining the f + given etale covering. As F is a sheaf on Xet , we have (iP F ) (U ) = F (U ) and hence (iP F )++ (U ) = F (U ). We thus have (iP F )# (U ) = F (U ). So r((iP F )# ) ∼ = F. Proposition 5.2.9. Let X be a quasi-compact quasi-separated scheme, and let Λ be a directed set. f (i) For any direct system (Fλ , φλµ )λ∈Λ of sheaves on Xet , the presheaf defined by U 7→ limλ∈Λ Fλ (U ) −→ f on Xet is a sheaf.

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on

f Xet

is a sheaf.

Proof. f (i) Let {Uα → U }α∈I be a covering in Xet . We have an exact sequence Y Y 0 → Fλ (U ) → Fλ (Uα ) → Fλ (Uα ×U Uβ ). α∈I

So the sequence

0 → lim Fλ (U ) → lim −→ −→ λ∈Λ

Y

λ∈Λ α∈I

α,β∈I

Fλ (Uα ) → lim −→

Y

λ∈Λ α,β∈I

Fλ (Uα ×U Uβ )

is exact. Since I is finite, the above sequence can be identified with the exact sequence Y Y 0 → lim Fλ (U ) → lim Fλ (Uα ) → lim Fλ (Uα ×U Uβ ). −→ −→ −→ λ∈Λ

α∈I λ∈Λ

α,β∈I λ∈Λ

f Xet .

Hence U 7→ limλ∈Λ Fλ (U ) is a sheaf on −→ (ii) follows from (i) and the fact that a direct sum over a set I is the direct limit of directs sums over finite subsets of I. Let C be a category on which fiber products exist. For each object U in C , suppose we specify a family TU whose elements are sets of morphisms in C of the form {Uα → U }α∈I . We call the elements in TU the coverings of U . We say that T is a Grothendieck topology on C if the following axioms hold: ∼ = (GT1) For any isomorphism V → U in C , the set {V → U } is a covering in TU . (GT2) Let {Uα → U }α∈I be a covering in TU . For any morphism V → U in C , the set {Uα ×U V → V }α∈I defined by base change is a covering in TV . (GT3) Let {Uα → U }α∈I be a covering in TU , and for each α ∈ I, let {Uαβ → Uα }β∈Iα be a covering in TUα . Then the set {Uαβ → U }α∈I,β∈Iα defined by taking composites is a covering in TU . A presheaf F on C is called a sheaf if for any U ∈ ob C and any covering {Uα → U }α∈I in TU , the sequence Y Y 0 → F (U ) → F (Uα ) → F (Uα ×U Uβ ) α∈I

α,β∈I

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is exact, where the homomorphisms in this sequence are defined by Y F (U ) → F (Uα ), s 7→ (s|Uα ), Y

α∈I

F (Uα ) →

Y

α,β∈I

α∈I

F (Uα ×U Uβ ),

(sα ) 7→ (sβ |Uα ×U Uβ − sα |Uα ×U Uβ ).

Many results in this section hold for sheaves on general Grothendieck topologies. 5.3

Stalks of Sheaves

([SGA 4] VIII 3, 4, 7.) Let Ω be a separably closed field. Schemes etale over Spec Ω are disjoint unions of copies of Spec Ω. So the functor F → Γ(Spec Ω, F ) defines an equivalence between the category of sheaves of sets on Spec Ω and the category of sets. Let Ω0 be a separably closed field containing Ω, and let u : Spec Ω0 → Spec Ω be the canonical morphism. Then the functor U 7→ U ⊗Ω Ω0 defines an equivalence between the category of etale Ω-schemes and the category of etale Ω0 -schemes. So the functor F 7→ u∗ F defines an equivalence between the category of sheaves on Spec Ω and the category of sheaves on Spec Ω0 . We have F (Spec Ω) ∼ = (u∗ F )(Spec Ω0 ). Let X be a scheme, let F be a sheaf on X, and let s → X be a geometric point of X, where s = Spec k(s) is the spectrum of a separably closed field k(s). For convenience, we often denote the geometric point simply by s. Let F |s be the inverse image of F on s. We define the stalk of F at s to be Γ(s, F |s ) and denote it by Fs . Let x ∈ X. Denote by x ¯ the geometric point Spec k(x) → X, where k(x) is a separable closure of k(x). If x is the image of s, then we have Fx¯ ∼ = Fs . Let f : X 0 → X be a morphism, let s0 be a geometric point of X 0 , and let f (s0 ) be the geometric point defined by the composite f

s0 → X 0 → X.

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Then for any sheaf F on X, we have (f ∗ F )s0 ∼ = Ff (s0 ) . Let X be a scheme and let s be a geometric point of X. Define a category Is as follows: Objects in Is are pairs (U, sU ), where U are objects in Xet , and sU : s → U are X-morphisms. For any two objects (U1 , sU1 ) and (U2 , sU2 ) in Is , a morphism f : (U1 , sU1 ) → (U2 , sU2 ) in Is is an Xmorphism f : U1 → U2 such that f sU1 = sU2 . We often denote an object (U, sU ) in Is simply by U . Objects in Is are called etale neighborhoods of s in X. Proposition 5.3.1. Let X be a scheme and let s → X be a geometric point of X. For any sheaf (resp. presheaf ) F on X, we have Fs ∼ = Proof.

lim −→

F (U )

(U,sU )∈ob Is◦

(resp. Fs# ∼ =

lim −→

F (U )).

(U,sU )∈ob Is◦

Note that for any presheaf G on s, we have Γ(s, G ) ∼ = Γ(s, G # ).

Let γ : s → X be the structure morphism. If F is a sheaf on X, we have Fs = Γ(s, γ ∗ F ) ∼ = Γ(s, γ P F ) =

lim −→

F (U ).

(U,sU )∈ob Is◦

If F is a presheaf on X, we have Fs# = Γ(s, γ ∗ (F # )) ∼ = Γ(s, (γ P F )# ) ∼ = Γ(s, γ P F ) =

lim −→

(U,sU )∈ob Is◦

F (U ).

Notation as above. Given an etale neighborhood (U, sU ) of s and any section t ∈ F (U ), we call the image of t in Fs ∼ F (U ) the = lim −→(U,sU )∈ob IS◦ germ of t at s. Lemma 5.3.2. Let X be a scheme and let F and G be sheaves on X. (i) Given a morphism u : F → G , if for any point x ∈ X, the map ux¯ : Fx¯ → Gx¯ induced by u is an isomorphism, then u is an isomorphism. (ii) Let ui : F → G (i = 1, 2) be two morphisms of sheaves. If for any x ∈ X, ui induce the same map Fx¯ → Gx¯ , then u1 = u2 . Proof. We give a proof of (i) and leave it for the reader to prove (ii). Let U be an object in Xet , and let si ∈ F (U ) (i = 1, 2) be two sections such that u(s1 ) = u(s2 ) in G (U ). By our assumption, for any point x0 ∈ U , s1 and s2 have the same image in Fx¯ , where x is the image of x0 in X. By 5.3.1,

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there exists an etale neighborhood Ux0 of x ¯0 in U such that s1 |Ux0 = s2 |Ux0 . 0 When x goes over the set of points of U , Ux0 form an etale covering of U . It follows that s1 = s2 . So u : F (U ) → G (U ) is injective. Let t ∈ G (U ). By our assumption, for any point x0 ∈ U , the image of t in Gx¯ lies in the image of Fx¯ → Gx¯ . So we can find an etale neighborhood Ux0 of x ¯0 and a section sx0 ∈ F (Ux0 ) such that u(sx0 ) = t|Ux0 . By the injectivity that we have shown, for any two points x01 , x02 ∈ U 0 , we have sx01 |Ux0 ×U Ux0 = sx02 |Ux0 ×U Ux0 . 1

2

1

2

So there exists a section s ∈ F (U ) such that s|Ux0 = sx0 for all x0 ∈ U . We then have u(s) = t. So u : F (U ) → G (U ) is surjective. Proposition 5.3.3. Let X be a scheme. A sequence F →G →H of morphisms of sheaves on X is exact if and only if for any geometric point s of X, the sequence Fs → Gs → Hs is exact. By 5.3.1, we have (ker (G → H ))s ∼ = ker (Gs → Hs ),

Proof.

(im (F → G ))s ∼ = im (Fs → Gs ).

This implies the “only if” part. If Fs → Gs → Hs is exact for all geometric points s in X, then by 5.3.2, the composite F → G → H is 0, and we have ∼ ker (G → H ). im (F → G ) = So F → G → H is exact.

Let A be a strictly henselian local ring and let s be the closed point of Spec A. By 2.3.10 (i) and 2.8.3 (vii), Spec A is a final object in Is◦ . So we have Fs ∼ = F (Spec A) for any sheaf F on Spec A. Let X be a scheme and let s → X be a geometric point of X. Consider the sheaf OXet on X. We have OXet (U ) = OU (U ) for any object U of Xet . So we have OXet ,s ∼ O (U ). = lim −→ ◦ U (U,sU )∈Is

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Let x be the image of s in X, and let OeX,s be the strict henselization of the local ring OX,x with respect to the separable closure of k(x) in k(s). Then we have OXet ,s ∼ = OeX,s .

es . We We call Spec OeX,s the strict localization of X at s and denote it by X es → X, and s → X can be factorized as have a canonical morphism X es → X. s→X

For any sheaf F on X, we have

es , F | e ). Fs ∼ = Γ(X Xs

Let s and s0 be two geometric points of X. A specialization morphism es0 → X es . When such a morphism exists, we say that is an X-morphism X 0 s is a generalization of s, and that s is a specialization of s0 .

Proposition 5.3.4. Let X be a scheme. (i) For any geometric point s0 in X and any etale X-scheme U , the canonical map es0 , U ) → HomX (s0 , U ) HomX (X

is bijective. (ii) Let s → X be another geometric point of X. Then the canonical map

is bijective.

es0 , X es ) → HomX (s0 , X es ) HomX (X

Proof. (i) Taking graphs of X-morphisms, we get the following one-to-one correspondences: es0 , U ) ∼ es0 , X es0 ×X U ), HomX (X = HomXes0 (X HomX (s0 , U ) ∼ = Homs0 (s0 , s0 ×X U ).

es0 -scheme W , that To prove our assertion, it suffices to show for any etale X the canonical map es0 , W ) → Homs0 (s0 , s0 × e W ) HomXe 0 (X X 0 s

s

is bijective. This follows from 2.3.10 (i) and 2.8.3 (vii). (ii) Let U0 be an affine open neighborhood of the image of s in X. Consider the full subcategory Js of Is consisting of those etale neighborhoods

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(U, sU ) of s such that U are affine and that the images of U in X are contained in U0 . Then Js is cofinal in Is . Moreover, Js is equivalent to a category whose objects form a set. We have

By 1.10.1 (i), we have

OeX,s ∼ =

lim −→

OU (U ).

(U,sU )∈ob Js◦

es ∼ X =

lim ←−

U

(U,sU )∈ob Js◦

in the category of schemes. So we have one-on-one correspondences es0 , X es ) ∼ HomX (X =

es ) ∼ HomX (s0 , X =

lim ←−

(U,sU )∈ob Js◦

lim ←−

es0 , U ), HomX (X HomX (s0 , U ).

(U,sU )∈ob Js◦

By (i), we have one-to-one correspondences es0 , U ) ∼ HomX (X = HomX (s0 , U ).

Our assertion follows.

Proposition 5.3.5. Let s and s0 be two geometric points in a scheme X, and let x and x0 be their images in X, respectively. Then s is a specialization of s0 if and only if x ∈ {x0 }. es ) is not Proof. If s is a specialization of s0 , then by 5.3.4, HomX (s0 , X 0 0 e empty. Let y be the image of an X-morphism s → Xs , and let y be the es . We have y ∈ {y 0 }. The images of y and y 0 in X are x closed point of X 0 and x , respectively. So we have x ∈ {x0 }. Conversely, suppose x ∈ {x0 }. Let (U, sU ) be an object in the category Js defined in the proof of 5.3.4 (ii). The image of U in X is open and contains x, and hence contains x0 . It follows that the set HomX (s0 , U ) of geometric points in U above s0 is not empty. This set es ) is finite. So lim(U,s )∈ob J HomX (s0 , U ) is not empty. Hence HomX (s0 , X ←− U s es0 → X es . So s is is not empty. By 5.3.4 (ii), there exists an X-morphism X a specialization of s0 .

Let s and s0 be two geometric points in a scheme X. Any specialization es0 → X es induces a homomorphism morphism X Fs → Fs0

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for any sheaf F on X. We call it the specialization map. This homomorphism can be described as follows: We have Fs ∼ lim F (U ), = −→ ◦ (U,sU )∈ob Is

Fs0 ∼ =

F (U 0 ).

lim −→

(U 0 ,s0U 0 )∈ob Is◦0

Given an object (V, sV ) in Is , the morphism sV : s → V induces a morphism es → V . Denote the composite X es0 → X es → V s0 → X 0 0 by sV . Then (V, sV ) is an object in Is0 . So we have canonical homomorphisms F (V ) F (V )

φ(V,sV )

→

φ(V,s0

→V

lim −→

F (U ),

(U,sU )∈ob Is◦ )

lim −→

F (U ).

(U 0 ,s0U 0 )∈ob Is◦0

For any section t ∈ F (V ), the homomorphism Fs → Fs0 maps φ(V,sV ) (t) to φ(V,s0V ) (t). Proposition 5.3.6. Let X be a scheme, s a geometric point in X, and s0 es . Denote the image of s0 in X also by s0 . Then X es0 a geometric point of X ∼ 0 es ) 0 of X es at s . is X-isomorphic to the strict localization (X s

es0 → X es . From its Proof. By 5.3.4, we have a canonical X-morphism X construction, one can show this is a flat morphism. Taking strict localizations at s0 of the following commutative diagram, es0 → X es , X ↓ . X we get the commutative diagram es0 → (X es )∼0 , X s id ↓ . es0 X

and hence a commutative diagram OeX,s0 ← OeXes ,s0 . id ↑ % OeX,s0

The horizontal arrow in the last diagram is thus surjective. It is also faithfully flat and hence injective. So it is an isomorphism.

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Proposition 5.3.7. Let f : X → Y be a finite morphism, y a point in Y , k(y) a separable closure of k(y), and F a sheaf on X. Then M (f∗ F )y¯ ∼ Fx¯ . = x∈X⊗OY k(y)

The functor f∗ is exact and faithful. In particular, a sequence F →G →H is exact if and only if f∗ F → f∗ G → f∗ H is exact. Proof. Let xi (i = 1, . . . , n) be all the points of X ⊗OY k(y). By 2.8.20, we have n a ex¯i . X ×Y Yey¯ ∼ X = i=1

The argument in the proof of 2.8.20 shows that there exists an etale mor`n phism U → Y such that U is affine, X ×Y U ∼ = i=1 Vi for some finite ex¯i . Replacing U -schemes Vi , Yey¯ → Y factors through U , and Vi ×U Yey¯ ∼ =X Y by U and X by each Vi , we are reduced to the case where Y is affine, ex¯ . X ⊗OY k(y) contains only one point x and X ×Y Yey¯ ∼ =X Fix an affine open neighborhood U0 of y in Y . By 5.3.4 (i), for any affine etale neighborhood (V, x¯V ) of x ¯ so that the image of V in Y is contained in U0 , the X-morphism x ¯V : Spec k(x) → V defines an X-morphism ex¯ → V g:X

ex¯ coincides with x so that the composite of g with Spec k(x) → X ¯V . Let Jy¯ be the category of affine etale neighborhoods of y¯ in U0 . We have Yey¯ ∼ = Spec (

lim −→

OU (U )).

(U,sU )∈ob Jy◦ ¯

By 1.10.9, there exist an affine etale neighborhood (U, sU ) of y¯ in Y and an X-morphism gU : X ×Y U → V so that the composite gU ex¯ ∼ X = X ×Y Yey¯ → X ×Y U → V

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ex¯ → V . The composite coincides with g : X

ex¯ ∼ Spec k(x) → X = X ×Y Yey¯ → X ×Y U

makes X ×Y U an etale neighborhood of x ¯ in X that dominates the etale neighborhood (V, x¯V ). It follows that Fx¯ ∼ =

lim −→

(U,sU )∈ob Jy◦ ¯

F (X ×Y U ).

But we have (f∗ F )y¯ ∼ =

lim −→

(U,sU )∈ob Jy◦ ¯

(f∗ F )(U ) ∼ =

lim −→

(U,sU )∈ob Jy◦ ¯

F (X ×Y U ).

So we have (f∗ F )y¯ ∼ = Fx¯ . This proves the first statement of the proposition. It implies that f∗ is exact, and that if f∗ F = 0, then F = 0. So f∗ is exact and faithful. Corollary 5.3.8. Let f : X → Y be an immersion. Then for any sheaf F on X, the canonical morphism f ∗ f∗ F → F is an isomorphism. Proof. The assertion is clear for any open immersion. If f is a closed immersion, we have (f ∗ f∗ F )x¯ ∼ = (f∗ F )f (¯x) ∼ = Fx¯ for any x ∈ X by 5.3.7. So f ∗ f∗ F ∼ = F for any immersion f .

Corollary 5.3.9. Consider a Cartesian diagram g0

X ×Y Y 0 → X f0 ↓ ↓f g Y 0 → Y.

Suppose f is finite. Then for any sheaf F on X, we have a canonical isomorphism ∼ =

g ∗ f∗ F → f∗0 g 0∗ F . Proof.

The composite of the canonical morphisms f∗ F

f∗ (adj)

→

f∗ g∗0 g 0∗ F ∼ = g∗ f∗0 g 0∗ F

induces a morphism g ∗ f∗ F → f∗0 g 0∗ F by adjunction. Using 5.3.7, one proves that this morphism induces isomorphisms on stalks. So it is an isomorphism.

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Corollary 5.3.10. Let f : X → Y be a finite surjective radiciel morphism. Then for any sheaf F on X and any sheaf G on Y , the canonical morphisms f ∗ f∗ F → F and G → f∗ f ∗ G are isomorphisms. So the functors f∗ and f ∗ define an equivalence between the category of sheaves on X and the category of sheaves on Y . Proof. By our assumption, for any y ∈ Y , X⊗OY k(y) consists of only one point. Using 5.3.7, one proves that the canonical morphisms f ∗ f∗ F → F and G → f∗ f ∗ G induce isomorphisms on stalks. 5.4

Recollement of Sheaves

Proposition 5.4.1. Let i : Y → X be a closed immersion, and let j : X − Y ,→ X be the complement. The functor i∗ : SY → SX is fully faithful, and a sheaf F on X is isomorphic to a sheaf of the form i∗ G for some sheaf G on Y if and only if j ∗ F = 0. Proof.

Let Gk (k = 1, 2) be two sheaves on Y . The canonical morphisms i∗ i∗ Gk → Gk

are isomorphisms. One can check the map Hom(G1 , G2 ) → Hom(i∗ G1 , i∗ G2 ) and the composite Hom(i∗ G1 , i∗ G2 ) → Hom(i∗ (i∗ G1 ), i∗ (i∗ G2 )) ∼ = Hom(G1 , G2 ) are inverses to each other. So i∗ is fully faithful. It is clear that j ∗ i∗ G = 0 for any sheaf G on Y . If F is a sheaf on X with the property j ∗ F = 0, then the canonical morphism F → i∗ i∗ F induces isomorphisms on stalks, and hence is an isomorphism. Let F be a sheaf on a scheme X. The smallest closed subset F of X such that F |X−F = 0 is called the support of F . Let U be an object in Xet , and let s ∈ F (U ) be a section. The support of s is the smallest closed subset F of U such that s|U−F = 0. Let V → U be an etale X-morphism. Then the support of s|V is the inverse image of the support of s in V . Let i : Y → X be a closed immersion, j : X − Y ,→ X the complement, F a sheaf on X, and K the kernel of the canonical morphism F → j∗ j ∗ F . Since j ∗ is exact, j ∗ K is the kernel of the morphism j ∗ F → j ∗ (j∗ j ∗ F ). But this morphism is clearly an isomorphism. So j ∗ K = 0. By 5.4.1, there

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exists a sheaf i! F on Y such that K ∼ = i∗ i! F . For any etale X-scheme ! U , sections in (i∗ i F )(U ) are those sections in F (U ) whose supports are contained in the inverse image of Y in U . For any sheaf G on Y , we have i! i∗ G ∼ = G . For any morphism φ : i∗ G → F , we have a commutative diagram i∗ G → F ↓ ↓ j∗ j ∗ (i∗ G ) → j∗ j ∗ F . As j∗ j ∗ (i∗ G ) = 0, the image of φ is contained in the kernel of F → j∗ j ∗ F . So we have Hom(i∗ G , F ) ∼ = Hom(i∗ G , i∗ i! F ) ∼ = Hom(G , i! F ). Hence i∗ is left adjoint to i! . Let H be a sheaf on X − Y . Define a presheaf P on X as follows: For any object V in Xet , if the image of V in X is not contained in X − Y , we let P(V ) = 0; otherwise, we regard V as an object of (X − Y )et and let P(V ) = H (V ). Define j! H = P # . We have j ∗ j! H ∼ = H . For any sheaf F on X, we have Hom(j! H , F ) ∼ = Hom(P, F ) ∼ = Hom(H , j ∗ F ). So j! is left adjoint to j ∗ . We summarize the above results as follows: Proposition 5.4.2. Let i : Y → X be a closed immersion, and let j : X − Y ,→ X be the complement. (i) (i∗ , i∗ ), (j! , j ∗ ), (i∗ , i! ) and (j ∗ , j∗ ) are pairs of adjoint functors. (ii) i∗ , i∗ , j ∗ , j! are exact, and j∗ , i! are left exact. (iii) i! i∗ ∼ = i∗ i∗ ∼ = id, ∗ ∼ ∗ j j! = j j∗ ∼ = id,

i∗ j! ∼ = i! j! ∼ = i! j∗ ∼ = 0, ∗ ∼ 0. j i∗ =

(iv) For any sheaf F on X, the following sequences are exact: 0 → i∗ i! F → F → j∗ j ∗ F ,

0 → j! j ∗ F → F → i∗ i∗ F → 0.

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Proposition 5.4.3. Let i : Y → X be a closed immersion, and let j : X − Y ,→ X be the complement. Define a category as follows: Objects in C are triples (G , H , φ), where G is a sheaf on Y , H is a sheaf on X − Y , and φ : G → i∗ j∗ H is a morphism. Given two objects (Gk , Hk , φk ) (k = 1, 2) in C , a morphism (ξ, η) : (G1 , H1 , φ1 ) → (G2 , H2 , φ2 ) in C consists of morphisms of sheaves ξ : G1 → G2 and η : H1 → H2 such that the following diagram commutes: φ1

G1 → i∗ j∗ H1 ξ ↓ ↓ i∗ j∗ (η) φ2

G2 → i∗ j∗ H2 .

Then the functor SX → C ,

F 7→ (i∗ F , j ∗ F , i∗ F → i∗ (j∗ j ∗ F ))

defines an equivalence of categories. Proof.

For any sheaf F on X, we have a commutative diagram F → i∗ i∗ F ↓ ↓ ∗ ∗ j∗ j F → i∗ i j∗ j ∗ F .

We claim that it is Cartesian. There exists a morphism F → i∗ i∗ F ×(i∗ i∗ j∗ j ∗ F ) j∗ j ∗ F whose composite with the projections are the canonical morphisms F → i∗ i∗ F and F → j∗ j ∗ F . Since i∗ and j ∗ are exact, we have i∗ (i∗ i∗ F ×(i∗ i∗ j∗ j ∗ F ) j∗ j ∗ F ) ∼ = i∗ i∗ i∗ F ×i∗ i∗ i∗ j∗ j ∗ F i∗ j∗ j ∗ F ∼ i∗ F ×i∗ j j ∗ F i∗ j∗ j ∗ F = ∗

∼ = i∗ F , ∼ j ∗ i∗ i∗ F ×j ∗ i i∗ j j ∗ F j ∗ j∗ j ∗ F j ∗ (i∗ i∗ F ×(i∗ i∗ j∗ j ∗ F ) j∗ j ∗ F ) = ∗ ∗ ∼ = 0 ×0 j ∗ F ∼ = j∗F .

It follows that the restriction of the morphism F → i∗ i∗ F ×(i∗ i∗ j∗ j ∗ F ) j∗ j ∗ F to Y and to X − Y are isomorphisms, and hence the morphism itself is an isomorphism. This proves our claim.

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Given two sheaves Fi (i = 1, 2) on X, the map Hom(F1 , F2 ) → Hom((i∗ F1 , j ∗ F1 , i∗ F1 → i∗ (j∗ j ∗ F1 )), (i∗ F2 , j ∗ F2 , i∗ F2 → i∗ (j∗ j ∗ F2 ))) is injective by 5.3.2 (ii). Let (ξ, η) : (i∗ F1 , j ∗ F1 , i∗ F1 → i∗ (j∗ j ∗ F1 )) → (i∗ F2 , j ∗ F2 , i∗ F2 → i∗ (j∗ j ∗ F2 )) be a morphism in C . By the claim above, there exists a morphism ϕ : F1 → F2 such that the following diagram commutes: i∗ i∗ F1 ← F1 → j∗ j ∗ F1 i∗ (ξ) ↓ ϕ ↓ ↓ j∗ (η) i∗ i∗ F2 ← F2 → j∗ j ∗ F2 . The morphism ϕ ∈ Hom(F1 , F2 ) is thus mapped to (ξ, η). This proves that our functor is fully faithful. For any object (G , H , φ) in C , define a sheaf F on X by the Cartesian diagram F → i∗ G ↓ ↓ i∗ φ j∗ H → i∗ i∗ j∗ H . Applying i∗ and j ∗ to this diagram, we get Cartesian diagrams i∗ F → G ↓ ↓φ id

i∗ j∗ H → i∗ j∗ H ,

j∗F → 0 ↓ ↓

H → 0.

It follows that i∗ F ∼ = G and j ∗ F ∼ = H . Moreover, the diagram i∗ F . ↓

∼ = G ↓φ id

i∗ j∗ j ∗ F → i∗ j∗ H → i∗ j∗ H . commutes. So (i∗ F , j ∗ F , i∗ F → i∗ j∗ j ∗ F ) ∼ = (G , H , φ). This proves that our functor is essentially surjective.

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The Functor f!

([SGA 4] XVII 6.1.) 0 Let f : X 0 → X be an etale morphism. We have a functor i : Xet → Xet 0 by regarding etale X -schemes as etale X-schemes. For any sheaf F on X, we have iP F ∼ = f ∗ F by 5.2.5 (iii). The functor iP has a left adjoint iP . For any object U in Xet , define a category KU as follows: Objects 0 of KU are pairs (U 0 , φ), where U 0 is an object in Xet , and φ : U → U 0 is an X-morphism. Given objects (Ui0 , φi ) (i = 1, 2) in KU , a morphism ξ : (U10 , φ1 ) → (U20 , φ2 ) in KU is an X 0 -morphism ξ : U10 → U20 such that ξφ1 = φ2 . The category KU◦ satisfies the conditions (I1) and (I2) in 2.7. For any presheaf F 0 on X 0 , we have (iP F 0 )(U ) = lim F 0 (U 0 ). −→ ◦ 0 (U ,φ)∈ob KU

Given an X-morphism ψ : U → X 0 , which is necessarily etale, denote by Uψ the etale X 0 -scheme U with the structure morphism ψ. The subcategory of KU◦ consisting of objects (Uψ , idU ) (ψ ∈ HomX (U, X 0 )) and the identity morphisms of these objects is a full cofinal subcategory. It follows that M F 0 (Uψ ). (iP F 0 )(U ) = ψ∈HomX (U,X 0 )

0

0

When F is a sheaf on X , we denote the sheaf (iP F 0 )# by f! F 0 . If f is an open immersion j, then f! coincides with the functor j! defined in 5.4. Proposition 5.5.1. Let f : X 0 → X be an etale morphism. (i) f! is left adjoint to f ∗ . (ii) Let g0

X 0 ×X Y → X 0 f0 ↓ ↓f g Y →X be a Cartesian diagram. For any sheaf F 0 on X 0 , we have a canonical isomorphism ∼ = f!0 g 0∗ F 0 → g ∗ f! F 0 . (iii) Let h : X 00 → X 0 be another etale morphism. We have (f h)! ∼ = f ! h! . (iv) For any point x in X, let k(x) be a separable closure of k(x). We have M Fx¯0 0 . (f! F 0 )x¯ ∼ = x0 ∈X 0 ⊗OX k(x)

In particular, the functor f! is exact and faithful.

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Proof. (i) For sheaves F 0 on X 0 and F on X, we have Hom(f! F 0 , F ) = Hom((iP F 0 )# , F ) ∼ = Hom(iP F 0 , F ) ∼ = Hom(F 0 , iP F ) ∼ = Hom(F 0 , f ∗ F ).

(ii) For any sheaf G on Y , since f is etale, we have a canonical isomorphism ∼ =

f ∗ g∗ G → g∗0 f 0∗ G . So we have Hom(g ∗ f! F 0 , G ) ∼ = Hom(F 0 , f ∗ g∗ G ) ∼ = Hom(F 0 , g 0 f 0∗ G ) ∗

∼ = Hom(f!0 g 0∗ F 0 , G ).

∼ g ∗ f! F 0 . It follows that f!0 g 0∗ F 0 = (iii) follows from (f h)∗ ∼ = h∗ f ∗ by adjunction. (iv) follows from (ii) by taking g to be the canonical morphism Spec k(x) → X. Suppose that f : X 0 → X is a separated etale morphism. Given a sheaf F on X, we construct a morphism 0

f! F 0 → f∗ F 0 by adjunction from a morphism F 0 → f ∗ f∗ F 0 as follows. Let φ : U 0 → X 0 be an etale morphism. We have (f ∗ f∗ F 0 )(U 0 ) ∼ = F 0 (U 0 ×X X 0 ).

Since f is separated, the graph Γφ : U 0 → U 0 × X X 0 of φ is a closed immersion. Since the projection U 0 ×X X 0 → U 0 is etale, Γφ is etale. So Γφ is an open and closed immersion by 2.3.8. We define a homomorphism F 0 (U 0 ) → F 0 (U 0 ×X X 0 )

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by mapping each section s ∈ F 0 (U 0 ) to the section of F 0 (U 0 ×X X 0 ) whose restriction via Γφ is s, and whose restriction to the complement of Γφ is 0. Recall that f! F 0 is the sheaf associated to the presheaf iP F 0 defined by M (iP F 0 )(U ) = F 0 (Uψ ) ψ∈HomX (U,X 0 )

for any etale X-scheme U . The morphism f! F 0 → f∗ F 0 is induced from the morphism of presheaves iP F 0 → f∗ F 0 defined by the homomorphism M

ψ∈HomX (U,X 0 )

F 0 (Uψ ) → F 0 (U ×X X 0 )

that maps each section s ∈ F 0 (Uψ ) to the section of F 0 (U ×X X 0 ) whose restriction via Γψ is s, and whose restriction to the complement of Γψ is 0. Proposition 5.5.2. Suppose that f : X 0 → X is an etale separated morphism of finite type. Then for any sheaf F 0 on X 0 , the canonical morphism f! F 0 → f∗ F 0

is injective. For any object U in Xet , a section s ∈ (f∗ F 0 )(U ) lies in the image of this morphism if any only if the support of s, considered as a section in F 0 (U ×X X 0 ), is proper over U . Here we say that a closed subset of U ×X X 0 is proper over U if any closed subscheme of U ×X X 0 , whose underlying topological space is the given closed subset, is proper over U . Proof. Note that the map ψ 7→ im Γψ defines a one-to-one correspondence from the set HomX (U, X 0 ) to the set of open and closed subsets of U ×X X 0 so that the restriction of the projection U ×X X 0 → U to them are isomorphisms. (Confer 2.3.10 (ii).) From the above description of the morphism iP F 0 → f∗ F 0 , one can verify that the morphism f! F 0 → f∗ F 0 is injective. It is clear that any section s ∈ F 0 (U ×X X 0 ) lying in the image of the morphism iP F 0 → f∗ F 0 of presheaves has proper support over U . Using 1.7.12, one deduces that any section s ∈ F 0 (U ×X X 0 ) lying in the image of the morphism f! F 0 → f∗ F 0 of sheaves has proper support over U. Let F be a closed subscheme of U ×X X 0 that is proper over U . Since f is necessarily quasi-finite, F is finite over U by [Fu (2006)] 2.5.12 or

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ex¯ be the strict localization of U at x¯. [EGA] III 4.4.2. For any x ∈ U , let U e e ex¯ ×U F as a closed subscheme of Then Ux¯ ×U F is finite over Ux¯ . Regard U 0 0 0 e ex¯ ×U F that lie above the Ux¯ ×X X . Let x¯1 , . . . , x ¯k be all the points in U ex¯ . By 2.8.14 (iii), there exist sections closed point of U ex¯ → U ex¯ ×X X 0 Γψi0 : U

(i = 1, . . . , k)

ex¯ ×X X 0 → U ex¯ such that Γψ0 map the closed point of of the projection U i 0 ex¯ to x U ¯i . The images of Γψi0 are open and closed connected subsets of ex¯ ×X X 0 , and they are disjoint. Moreover, we have U ex¯ ×U F ⊂ U

k [

im Γψi0 .

i=1

S ex¯ ×U F − k im Γψ0 is a closed subset of U ex¯ ×U F . Since F Indeed, U i=1 i Sk ex¯ ×U F − e ¯ is closed. If 0 is proper over U , the image of U i=1 im Γψi in Ux ex¯ . This is it is nonempty, then this image contains the closed point of U 0 ex¯ in U ex¯ ×U F impossible since all the preimages xi of the closed point of U Sk ex¯ ×U F − Sk im Γψ0 is empty. By 1.10.1 (iv) lie in i=1 im Γψi0 . Hence U i=1 i and 1.10.9, there exist an etale neighborhood Ux¯ of x ¯ in U and sections Γψi : Ux¯ → Ux¯ ×X X 0

(i = 1, . . . , k)

of the projection Ux¯ ×X X 0 → Ux¯ such that im Γψi are disjoint open and closed subsets of Ux¯ ×X X 0 , and Ux¯ ×U F ⊂

k [

im Γψi .

i=1

Let ψi : Ux¯ → X 0 be the composite Γψ

Ux¯ →i Ux¯ ×X X 0 → X 0 . For any section s ∈ F 0 (U ×X X 0 ) with the property s|U×X X 0 −F = 0, note that s|Ux¯ ×X X 0 lies in the image of the homomorphism M F 0 (Ux¯,ψ ) → F 0 (Ux¯ ×X X 0 ) ψ∈HomX (Ux¯ ,X 0 )

described before 5.5.2. As Ux¯ (x ∈ U ) form an etale covering of U , s lies in the image of the morphism of sheaves f! F 0 → f∗ F 0 .

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Let f : X 0 → X be a separated morphism of finite type and let F 0 be a sheaf on X 0 . Define a presheaf f! F 0 on X so that for any object U in Xet , we have (f! F 0 )(U ) = {s|s ∈ F 0 (U ×X X 0 ) and the support of s is proper over U }. Using 1.7.12, one can show that f! F 0 is a sheaf. We have a canonical monomorphism f! F ,→ f∗ F . If f is etale, then f! coincides with the functor defined at the beginning of this section by 5.5.2. If f is proper, we have f! = f∗ . Proposition 5.5.3. Let f : X 0 → X and f 0 : X 00 → X 0 be two separated morphisms of finite type and let g = f f 0 . Then we have an isomorphism of functors g! ∼ = f! f!0 such that the following diagram commutes: g! ∼ = f! f!0 ↓ ↓ g∗ ∼ f = ∗ f∗0 . Proof. Let U be an object in Xet and let F be a sheaf on X 00 . We need to show that under the identification (g∗ F )(U ) ∼ = (f∗ f∗0 F )(U ), the subset (g! F )(U ) of (g∗ F )(U ) is identified with the subset (f! f!0 F )(U ) of (f∗ f∗0 F )(U ). Let s ∈ F (U ×X X 00 ) be a section lying in (f! f!0 F )(U ). Then s lies in (f∗ f!0 F )(U ) = (f!0 F )(U ×X X 0 ). So the support of s is proper over U ×X X 0 . On the other hand, s lies in (f! f∗0 F )(U ). So if we consider s as a section of (f∗0 F )(U ×X X 0 ), its support is proper over U . The support of s considered as a section of (f∗0 F )(U ×X X 0 ) is the closure of the image of the support of s under the morphism U ×X X 00 → U ×X X 0 . One deduces from these facts that the support of s is proper over U . So s ∈ (g! F )(U ). Let s ∈ F (U ×X X 00 ) be a section lying in (g! F )(U ). Then the support of s is proper over U . Since U ×X X 0 → U is separated, the support of s is also proper over U ×X X 0 . Hence s lies in (f!0 F )(U ×X X 0 ). Moreover, the image of the support of s in U ×X X 0 is closed. So the support of s considered as a section of (f!0 F )(U ×X X 0 ) coincides with image of the support of s under the morphism U ×X X 00 → U ×X X 0 . Since the support of s is proper over U , its image in U ×X X 0 is also proper over U . It follows that s ∈ (f! f!0 F )(U ).

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Etale Cohomology

([SGA 4] XVII 4.2.2–4.2.6.) Let X be a scheme, and let F : SX → A be a left exact functor from the category SX of sheaves of abelian groups on X to an abelian category A . By 5.2.4 (i), we can define the right derived functors Ri F (i ≥ 0) of F . For any object U in Xet , the functor F 7→ F (U ) is left exact. Its right derived functors are denoted by F 7→ H i (U, F ), or F 7→ Ri Γ(U, F ). We call H i (X, F ) the etale cohomology groups of F . Note that we have H i (U, F ) ∼ = H i (U, F |U ). Let f : X 0 → X be a morphism of schemes. The functor F 0 7→ f∗ F 0 is left exact. Its right derived functors are denoted by F 0 7→ Ri f∗ F 0 . We call Ri f∗ F 0 the higher direct images of F 0 . Each Ri f∗ F 0 is the sheaf on X associated to the presheaf U 7→ H i (U ×X X 0 , F 0 ) for any U ∈ ob Xet . Let A be a ring. The category of sheaves of A-modules on X has enough injective objects. For any sheaf of A-modules G on X. The functors F 7→ HomA (G , F ),

F 7→ H omA (G , F )

are left exact. Their derived functors are denoted by F 7→ ExtiA (G , F ),

F 7→ E xtiA (G , F )

respectively. We often denote ExtiA (F , G ) and E xtiA (F , G ) by Exti (F , G ) and E xti (F , G ), respectively. Lemma 5.6.1. Let F be a sheaf on a scheme X and let H i (F ) be the presheaf defined by H i (F )(U ) = H i (U, F ) for any object U in Xet . Then H i (F )+ = 0 for all i ≥ 1. Proof. Let I · be an injective resolution of F . Then H i (F )(U ) is the i-th cohomology group of the complex I · (U ). So H i (F )# is the i-th cohomology sheaf of the complex I · . Hence H i (F )# = 0 for all i ≥ 1. Since H i (F )+ are subsheaves of H i (F )# , we have H i (F )+ = 0 for all i ≥ 1. Proposition 5.6.2. Let X be a scheme and let U = {Uα → U }α∈I be an etale covering in Xet . For any sheaf F on X, we have biregular spectral sequences ˇ i (U, H j (F )) ⇒ H i+j (U, F ), E2ij = H ˇ i (U, H j (F )) ⇒ H i+j (U, F ), E2ij = H

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where ˇ i (U, −) = H

ˇ i (U, −). lim H −→

U∈ob JU

(See the discussion after 5.2.1 for the definition of JU ). Proof. If I is an injective sheaf, then one can show that it is an injective ˇ j (U, I ) = 0 for all j ≥ 1 object in the category of presheaves. So we have H · by 5.1.1. Let I be an injective resolution of F . Consider the bicomplex C · (U, I · ). We have Γ(U, I i ) if j = 0, j · i j i ˇ H (C (U, I )) = H (U, I ) = 0 if j ≥ 1. So the spectral sequence i E2ij = HII HIj (C · (U, I · )) ⇒ H i+j (C · (U, I · ))

degenerates and we have H i (C · (U, I · )) ∼ = H i (Γ(U, I · )) ∼ = H i (U, F ). The spectral sequence j E2ij = HIi HII (C · (U, I · )) ⇒ H i+j (C · (U, I · ))

can be identified with ˇ i (U, H j (F )) ⇒ H i+j (U, F ). E2ij = H Taking direct limit, we get the spectral sequence ˇ i (U, H j (F )) ⇒ H i+j (U, F ). E2ij = H ˇ 1 (U, F ) Corollary 5.6.3. We have a canonical isomorphism H 1 2 2 ˇ H (U, F ) and a canonical monomorphism H (U, F ) ,→ H (U, F ). Proof.

∼ =

We have ˇ 0 (U, H j (F )) ∼ H = H j (F )+ (U ) = 0

for all j ≥ 1 by 5.6.1. We then use the exact sequence 0 → E210 → H 1 (U, F ) → E201 → E220 → H 2 (U, F ) in [Fu (2006)] 2.2.3 deduced from the second spectral sequence in 5.6.2.

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A sheaf F on a scheme X is called flasque if for any etale covering ˇ 1 (U, F ) = 0. By 5.6.3, we have U = {Uα → U }α∈I in Xet , we have H ˇ 1 (U F ) ∼ H 1 (U, F ) ∼ =H =

lim −→

ˇ 1 (U, F ) = 0 H

U∈ob JU

for any U ∈ ob Xet and any flasque sheaf F on X. Proposition 5.6.4. Let 0 → F 0 → F → F 00 → 0 be an exact sequence of sheaves on a scheme X. (i) If F 0 is flasque, then the sequence is exact in the category of presheaves. (ii) If F 0 and F are flasque, then F 00 is flasque. (iii) If the sequence splits and F is flasque, then F 0 and F 00 are flasque. (iv) Injective sheaves are flasque. Proof. We prove (i) and leave the rest to the reader. For any object U in Xet , we have a long exact sequence 0 → F 0 (U ) → F (U ) → F 00 (U ) → H 1 (U, F 0 ) → · · · .

If F 0 is flasque, then H 1 (U, F 0 ) = 0 and hence the sequence 0 → F 0 (U ) → F (U ) → F 00 (U ) → 0 is exact.

Proposition 5.6.5. Let f : X 0 → X be a morphism of schemes. (i) If F 0 is a flasque sheaf on X 0 , then f∗ F 0 is flasque. (ii) Let 0 → F 0 → F → F 00 → 0

be an exact sequence of sheaves on X 0 . If F 0 is flasque, then 0 → f∗ F 0 → f∗ F → f∗ F 00 → 0 is exact. Proof. (i) Let U = {Uα → U } be an etale covering in Xet . Then f ∗ U = 0 {Uα ×X X 0 → U ×X X 0 } is an etale covering in Xet . We have ˇ i (U, f∗ F 0 ) ∼ ˇ i (f ∗ U, F 0 ). H =H

ˇ 1 (f ∗ U, F 0 ) = 0. So H ˇ 1 (U, f∗ F 0 ) = 0. Hence If F 0 is flasque, we have H 0 f∗ F is flasque.

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(ii) By 5.6.4 (i), the sequence 0 → F 0 → F → F 00 → 0 is exact in the category of presheaves. So 0 → f∗ F 0 → f∗ F → f∗ F 00 → 0 is exact in the category of presheaves and hence exact in the category of sheaves. Corollary 5.6.6. (i) If F is a flasque sheaf on a scheme X, then H i (U, F ) = 0 for all i ≥ 1 and all objects U in Xet . So we can use flasque resolutions to calculate H i (U, −). (ii) Let f : X 0 → X be a morphism of schemes. For all flasque sheaves 0 F on X 0 , we have Ri f∗ F 0 = 0 for all i ≥ 1. So we can use flasque resolutions to calculate Ri f∗ . Proof. We prove (i) and leave it for the reader to prove (ii). Let I · be an injective resolution of F , and let Z i = ker (I i → I i+1 ). Then we have short exact sequences 0 → F → I 0 → Z 1 → 0, 0 → Z 1 → I 1 → Z 2 → 0, .. . Using 5.6.4 (ii) and (iv), one can show that F = Z 0 , Z 1 , · · · are flasque. By 5.6.4 (i), the following sequences are exact: 0 → F (U ) → I 0 (U ) → Z 1 (U ) → 0, 0 → Z 1 (U ) → I 1 (U ) → Z 2 (U ) → 0, .. . So the sequence I 0 (U ) → I 1 (U ) → · · · is exact. Hence H i (U, F ) = 0 for all i ≥ 1.

Proposition 5.6.7. Let F be a sheaf on a scheme X. The following conditions are equivalent: (i) For any object U in Xet , we have H i (U, F ) = 0 for all i ≥ 1. ˇ i (U, F ) = 0 for all i ≥ 1. (ii) For any object U in Xet , we have H (iii) For any etale covering U = {Uα → U }α∈I in Xet , we have ˇ i (U, F ) = 0 for all i ≥ 1. H (iv) F is flasque.

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Proof. (i)⇒(iii) Suppose H j (U, F ) = 0 for all U ∈ ob Xet and all j ≥ 1. Then j H (F ) = 0 for all j ≥ 1. The spectral sequence ˇ i (U, H j (F )) ⇒ H i+j (U, F ) E ij = H 2

degenerates and we have ˇ i (U, H 0 (F )) ∼ ˇ i (U, F ) ∼ H = H i (U, F ) = 0 =H

for all i ≥ 1. (iii)⇒(ii) is clear. (ii)⇒(i) Let us prove H j (F ) = 0 for all j ≥ 1 by induction on j. For any object V in Xet , by 5.6.3, we have ˇ 1 (V, F ) = 0. H 1 (F )(V ) = H 1 (V, F ) ∼ =H Suppose we have shown H j (F ) = 0 for all 1 ≤ j ≤ n. Consider the spectral sequence ˇ i (U, H j (F )) ⇒ H i+j (U, F ). E ij = H 2

We have

E20j = H j (F )+ (U ) = 0 for all j ≥ 1 by 5.6.1 and E2ij = 0 for all 1 ≤ j ≤ n. This implies that ˇ n+1 (U, F ) ∼ H = H n+1 (U, F ) by [Fu (2006)] 2.2.3. So H n+1 (U, F ) = 0 and hence H n+1 (F ) = 0. (iii)⇒(iv) is clear. (iv)⇒(i) follows from 5.6.6 (i).

Let X be a scheme and let A be a ring. Injective objects in the category of sheaves of A-modules on X are called injective sheaves of A-modules. Using the same argument as in the proof of 5.1.1 by replacing Z by A, one can show that injective sheaves of A-modules are flasque. So for any sheaf F of A-modules on X, we can also use resolutions of F by injective sheaves of A-modules to calculate H i (U, F ) and Ri f∗ F for any U ∈ ob Xet and any morphism f : X → Y . We can define A-module structures on the groups H i (U, F ) and the sheaves Ri f∗ F . We say that a sheaf F of A-modules on X is flat if the stalks of F at all geometric points of X are flat A-modules. This is equivalent to saying that the functor F ⊗A − is exact in the category of sheaves of A-modules. Lemma 5.6.8. Let X be a scheme, A a ring, F a sheaf of A-modules, and I an injective sheaf of A-modules. If F is flat, then H omA (F , I ) is injective. In general, H omA (F , I ) is flasque.

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Proof.

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We have HomA (−, H omA (F , I )) ∼ = HomA (F ⊗A −, I ).

If F is flat and I is injective, then the functor HomA (F ⊗A −, I ) is exact. So the functor HomA (−, H omA (F , I )) is exact, and hence H omA (F , I ) is an injective sheaf of A-modules. This implies that H omA (F , I ) is flasque. For any etale morphism f : U → X, let AU = f! A, where A on the right-hand side denotes the constant sheaf on U associated to A, that is, the sheaf associated to the constant presheaf V 7→ A for any etale U -scheme V . Let U = {Uα → U } be an etale covering of U . As in the proof of 5.1.1, we have an exact sequence M δ−1 M δ0 δ1 0 ← AU ← AUα0 ← AUα0 α1 ← ··· . α0

α0 ,α1

For convenience, we write the above sequence as A·U → AU → 0. Since I is an injective sheaf of A-modules, the sequence 0 → H omA (AU , I ) → H omA (A·U , I ) is exact. Since AU and AUα0 ...αi are flat sheaves, H omA (A·U , I ) is an injective resolution of the injective sheaf H omA (AU , I ) in the category of sheaves of A-modules. So we have H i (HomA (F , H omA (A·U , I ))) ∼ = Exti (F , H omA (AU , I )) = 0 for all i ≥ 1. On the other hand, we have H i (HomA (F , H omA (A·U , I ))) ∼ = H i (HomA (A·U , H omA (F , I ))) ∼ = H i (C · (U, H omA (F , I ))) ∼ ˇ i (U, H omA (F , I )). =H

ˇ i (U, H omA (F , I )) = 0 for all i ≥ 1. Hence H omA (F , I ) So we have H is flasque. Corollary 5.6.9. (i) Let f : X 0 → X be a morphism of schemes and let F 0 be a sheaf on 0 X . We have a biregular spectral sequence E2pq = H i (X, Rj f∗ F 0 ) ⇒ H i+j (X 0 , F 0 ).

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(ii) Let f 0 : X 00 → X 0 and f : X 0 → X be morphisms of schemes and let F 00 be a sheaf on X 00 . We have a biregular spectral sequence E2ij = Ri f∗ Rj f∗0 F 00 ⇒ Ri+j (f f 0 )∗ F 00 . (iii) Let F and G be sheaves of A-modules on a scheme X. We have a biregular spectral sequence E2ij = H i (X, E xtiA (F , G )) ⇒ Exti+j A (F , G ). Proof. This follows from 5.6.5, 5.6.6, 5.6.8, [Fu (2006)] 2.2.10 or [Grothendieck (1957)] 2.4.1. Let X be a scheme and let {Uα → X} be an etale covering of X. If I is an injective sheaf, the proof of 5.6.8 shows that M M 0 → H om( ZUα0 , I ) → H om( ZUα0 α1 , I ) → · · · α0

α0 ,α1

is an injective resolution of H om(ZX , I ). Denote by πα0 ...αn the canonical morphism Uα0 ...αn = Uα0 ×X · · · ×X Uαn → X. This shows that Y Y πα0 α1 ∗ πα∗ 0 α1 I → · · · πα0 ∗ πα∗ 0 I → 0→ α0

α0 ,α1

is an injective resolution of I . Let f : X → Y be a morphism, let F be a sheaf on X, a let I · be an injective resolution of F . Consider the bicomplex .. .. . . ↑ ↑ Q Q 0 → α0 πα0 ∗ πα∗ 0 I1 → α0 ,α1 πα0 α1 ∗ πα∗ 0 α1 I1 → · · · ↑ ↑ Q Q 0 → α0 πα0 ∗ πα∗ 0 I0 → α0 ,α1 πα0 α1 ∗ πα∗ 0 α1 I0 → · · · ↑ ↑ 0 0 Applying f∗ to this bicomplex and analyzing the spectral sequences associated to the resulting bicomplex, we get the following. Proposition 5.6.10. Notation as above. We have a biregular spectral sequence E2ij ⇒ Ri+j f∗ F , where E2ij is the i-th cohomology sheaf of the complex Y Y 0→ Rj (f πα0 )∗ πα∗ 0 F → Rj (f πα0 α1 )∗ πα∗ 0 α1 F · · · . α0

α0 ,α1

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Let i : Y → X be a closed immersion and let j : U = X − Y ,→ X be its complement. For any sheaf F on X, let ΓY (X, F ) = {s ∈ Γ(X, F )| the support of s is contained in Y }. The functor ΓY (X, −) is left exact. Denote its right derived functors by HYq (X, −) or Rq ΓY (X, −). Let Rq i! be the right derived functors of i! . Proposition 5.6.11. Notation as above. (i) We have a long exact sequence · · · → HYq (X, F ) → H q (X, F ) → H q (U, F ) → · · · . (ii) We have an exact sequence 0 → i∗ i! F → F → j∗ j ∗ F → i∗ R1 i! F → 0 and isomorphisms Rq j∗ (j ∗ F ) ∼ = i∗ (Rq+1 i! F )

(q ≥ 1).

(iii) We have a biregular spectral sequence E2pq = H p (Y, Rq i! F ) ⇒ HYp+q (X, F ). Proof. Denote the constant sheaves on U and on X associated to Z both by Z. Note that j! Z is a subsheaf of Z. For any injective sheaf I on X, the canonical homomorphism Hom(Z, I ) → Hom(j! Z, I ) is surjective. So the restriction Γ(X, I ) → Γ(U, I ) is surjective. Similarly, for any etale X-scheme V , the restriction I (V ) → I (U ×X V ) is surjective. So the morphism I → j∗ j ∗ I

is surjective. Let I · be an injective resolution of F . We have a short exact sequence of complexes 0 → ΓY (X, I · ) → Γ(X, I · ) → Γ(U, I · ) → 0. Taking the long exact sequence of cohomology groups associated to this short exact sequence, we get (i). We have a short exact sequence of complexes 0 → i∗ i! I · → I · → j∗ j ∗ I · → 0.

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Taking the long exact sequence of cohomology sheaves associated to this short exact sequence, we get (ii). Since the functor i∗ is exact and is left adjoint to i! , the functor i! maps injective sheaves to injective sheaves. Moreover, we have Γ(Y, i! −) = ΓY (X, −).

By [Fu (2006)] 2.2.10 or [Grothendieck (1957)] 2.4.1, we have a biregular spectral sequence E2pq = H p (Y, Rq i! F ) ⇒ HYp+q (X, F ).

Proposition 5.6.12 (Excision Theorem). Let f : X 0 → X be an etale morphism, Y → X a closed immersion, Y 0 = X 0 ×X Y , and F a sheaf on X. Suppose that the projection Y 0 → Y is an isomorphism. Then we have q H q (X, F ) ∼ = H 0 (X 0 , f ∗ F ) Y

Y

for all q. Proof. Since f! is left adjoint to f ∗ and is exact, f ∗ maps injective sheaves to injective sheaves. Moreover f ∗ is exact. To prove our assertion, it suffices to prove that the canonical homomorphism ΓY (X, F ) → ΓY 0 (X 0 , f ∗ F )

is bijective for any sheaf F on X. Suppose s ∈ ΓY (X, F ) and s|X 0 = 0. We have s|X−Y = 0. Since X − Y ,→ X and X 0 → X form an etale covering of X, we have s = 0 and hence the above homomorphism is injective. Suppose s0 ∈ ΓY 0 (X 0 , f ∗ F ). Let us prove that (s0 , 0) lies in the kernel of the homomorphism d

F (X 0 ) × F (X − Y ) →0 F (X 0 ×X X 0 ) × F (X 0 ×X (X − Y )) × F ((X − Y ) ×X X 0 ) × F ((X − Y ) ×X (X − Y ))

ˇ in the Cech complex for the etale covering {X − Y ,→ X, X 0 → X}, and hence there exists s ∈ Γ(X, F ) such that s|X 0 = s0 and s|X−Y = 0. This shows that the above homomorphism is surjective. The nontrivial part is to verify that s0 is mapped to the same section under the restrictions p∗i : F (X 0 ) → F (X 0 ×X X 0 ) (i = 1, 2)

defined by the projections pi : X 0 ×X X 0 → X 0 . Note that X 0 ×X X 0 can be covered by the open subscheme (X 0 − Y 0 ) ×(X−Y ) (X 0 − Y 0 ) and the closed subscheme Y 0 ×Y Y 0 . It is clear that p∗i (s0 ) vanish on (X 0 −Y 0 )×(X−Y ) (X 0 − Y 0 ). Since Y 0 → Y is an isomorphism, the two projections Y 0 ×Y Y 0 → Y 0 are the same. So p∗1 (s0 ) and p∗2 (s0 ) have the same germ at every geometric point of Y 0 ×Y Y 0 . We thus have p∗1 (s0 ) = p∗2 (s0 ).

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Let X be a scheme and let P = {γs : s → X} be a set of geometric points of X such that for every point x in X, there exists a geometric point γs in P with image x, where s are spectra of separably closed fields. For any sheaf F on X, let Y C 0 (F ) = γs∗ γs∗ F . s∈P

0

The canonical morphism F → C (F ) is injective. Let C −1 (F ) = F and let d−1 : C −1 (F ) → C 0 (F ) be this monomorphism. Suppose we have defined an exact sequence d−1

d0

dn−1

0 → F → C 0 (F ) → · · · → C n (F ) for some n ≥ 0. We define

dn−1

C n+1 (F ) = C 0 (coker (C n−1 (F ) → C n (F ))) and define dn+1 : C n (F ) → C n+1 (F ) to be the composite dn−1

dn−1

C n (F ) → coker (C n−1 (F ) → C n (F )) → C 0 (coker (C n−1 (F ) → C n (F ))). Then we get a resolution d0

dn−1

dn

dn+1

0 → C 0 (F ) → · · · → C n (F ) → C n+1 (F ) → · · · of F . We denote it by C · (F ) and call it the Godement resolution of F . Let A be a ring, di

K · = (· · · → K i → K i+1 → · · · ) an acyclic complex of A-modules, and Z i = ker di . Then we have short exact sequences 0 → Z i → K i → Z i+1 → 0. We say K · is split if all these short exact sequences are split. Proposition 5.6.13. dn−1

(i) The functors F 7→ C n (F ) and F 7→ im (C n−1 (F ) → C n (F )) are exact for all n ≥ 0. (ii) C n (F ) (n ≥ 0) are flasque. (iii) The stalk of the complex 0 → F → C · (F ) at every geometric point s in P is split.

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Proof. (i) For any object U in Xet and any geometric point s in P , s ×X U is a disjoint union of copies of s, and we have (γs∗ γs∗ F )(U ) = (γs∗ F )(s ×X U ). It follows that γs∗ γs∗ is an exact functor and hence C 0 (−) is an exact functor. Suppose that 0→F →G →H →0 is an exact sequence. Applying the snake lemma to the commutative diagram 0→

F ↓

→

G ↓

→

H ↓

→ 0

0 → C 0 (F ) → C 0 (G ) → C 0 (H ) → 0, and using the fact that the vertical arrows are monomorphisms, we get an exact sequence 0→coker (F →C 0 (F ))→coker (G →C 0 (G ))→coker (H →C 0 (H ))→0.

Suppose we have proved that 0→C n (F )→C n (G )→C n (H )→0, 0→coker (C n−1 (F )→C n (F ))→coker (C n−1 (G )→C n (G ))→coker (C n−1 (H )→C n (H ))→0

are exact. Applying C 0 (−) to the second exact sequence, we get an exact sequence 0 → C n+1 (F ) → C n+1 (G ) → C n+1 (H ) → 0. Applying the snake lemma to the commutative diagram 0→ coker(C n−1 (F)→C n (F)) → coker(C n−1 (G )→C n (G )) → coker(C n−1 (H )→C n (H )) → 0 ↓ ↓ ↓ 0→

C n+1 (F)

→

C n+1 (G )

→

C n+1 (H )

→0

and using the fact that vertical arrows are monomorphisms, we get an exact sequence 0→coker (C n (F )→C n+1 (F ))→coker (C n (G )→C n+1 (G ))→coker (C n (H )→C n+1 (H ))→0.

So the functors F → 7 C n (F ) and F 7→ coker (C n−1 (F ) → C n (F )) are exact for all n ≥ 0. We have coker (C n−1 (F ) → C n (F )) ∼ = im (C n (F ) → C n+1 (F )).

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So the functors F 7→ im (C n (F ) → C n+1 (F )) are exact for all n ≥ 0. For n = −1, this functor can be identified with the identity functor and hence is exact. (ii) It is enough to show that C 0 (F ) is flasque. For any s ∈ P , the functor Γ(s, −) is exact and hence H i (s, −) = 0 for all i ≥ 1. So any sheaf on s is flasque. By 5.6.5 (i), γs∗ γs∗ F is flasque. This implies that Q C 0 (F ) = s∈P γs∗ γs∗ F is flasque. (iii) We first prove the following fact: For any morphism f : X → Y and any sheaf F on Y , the diagram f ∗ F → f ∗ (f∗ f ∗ F ) idk kid f ∗ F ← (f ∗ f∗ )f ∗ F commutes, where the horizontal arrows are induced by the canonical moradj adj phisms id → f∗ f ∗ and f ∗ f∗ → id, respectively. Indeed, the composite f ∗ F → f ∗ (f∗ f ∗ F ) = (f ∗ f∗ )f ∗ F → f ∗ F is induced by the composite adj

id

F → f∗ f ∗ F → f∗ f ∗ F by adjunction. It is clear that id : f ∗ F → f ∗ F is also induced by the adj morphism F → f∗ f ∗ F by adjunction. So the above diagram commutes. Let t ∈ P . Consider the morphism γt∗ C 0 (F ) → γt∗ F defined by composing the projection Y γt∗ C 0 (F ) = γt∗ γs∗ γs∗ F → γt∗ (γt∗ γt∗ F ) s∈P

with the canonical morphism γt∗ (γt∗ γt∗ F ) = (γt∗ γt∗ )γt∗ F → γt∗ F . By the above discussion, this morphism is a left inverse of the stalk at t of the morphism F → C 0 (F ). It follows that the short exact sequence 0 → Ft → C 0 (F )t → ker (C 1 (F ) → C 2 (F ))t → 0 is split. Applying this result to coker (C n−2 (F ) → C n−1 (F )), we see that the short exact sequences 0→ker(C n (F )→C n+1 (F ))t →C n (F )t →ker (C n+1 (F )→C n+2 (F ))t →0

are split for all n.

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Let S be a scheme. For every s ∈ S, fix an algebraic closure k(s) of the residue field k(s). For every S-scheme X locally of finite type, let [ PX = HomS (Spec k(s), X) s∈S

be the set of geometric points above the geometric points Spec k(s) → S in S. For every sheaf F on X, we define a complex of sheaves C · (F ) and a morphism F → C 0 (F ) as before using PX instead of P . Then C · (F ) is resolution of F (see the discussion below) and 5.6.13 still holds. We also call C · (F ) the Godement resolution of F . This resolution has the advantage that it is functorial, that is, if f : X 0 → X is an S-morphism between S-schemes locally of finite type, then for any sheaf F on X, we have a canonical morphism of complexes C · (F ) → f∗ C · (f ∗ F ). Let us prove that the morphism F → C 0 (F ) is injective. Let π : U → X be an object in Xet , and let ξ ∈ F (U ) such that its image in C 0 (F )(U ) is 0. We need to show ξ = 0. Let f : X → S be the structure morphism. For every u ∈ U such that u is closed in the fiber (f π)−1 (f π(u)), k(u) is a finite extension of k(f (π(u))). So there exists an S-morphism Spec k(f π(u)) → U whose image is u. Composing this morphism with π, we get a geometric point in PX . Since the image of ξ in Y C 0 (F )(U ) = (γt∗ γt∗ F )(U ) t∈PX

is 0, the germ of ξ at the geometric point u ¯ is 0. So there exists an open S neighborhood Vu of u in U such that ξ|Vu = 0. Let V = u∈U Vu , where u goes over all points in U which are closed in the fibers (f π)−1 (f π(u)). Then V is an open subset of U and ξ|V = 0. To prove ξ = 0, it suffices to show V = U . For any s ∈ S, (f π)−1 (s) is a k(s)-scheme locally of finite type, and V ∩ (f π)−1 (s) is an open subset containing all the closed points of (f π)−1 (s). It follows that V ∩ (f π)−1 (s) = (f π)−1 (s). 5.7

Calculation of Etale Cohomology

([SGA 4] VII 4, VIII 1, 2, [SGA 4 21 ] Arcata I 5, III 1.) Proposition 5.7.1. Let f : X → Y be a finite surjective radiciel morphism. For any sheaf F on Y , we have H i (Y, F ) ∼ = H i (X, f ∗ F ).

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Proof.

Use 5.3.10.

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Corollary 5.7.2. (i) Let X be a scheme. Then H i (X, F ) ∼ = H i (Xred , F |Xred ) for any sheaf F on X. (ii) Let X be a scheme over a field K and let L be a finite purely inseparable extension of K. Then H i (X, F ) ∼ = H i (X ⊗K L, F |X⊗K L ) for any sheaf F on X. Proposition 5.7.3. Let A be a strict henselian local ring. For any sheaf F on X = Spec A, we have H i (X, F ) = 0 for all i ≥ 1. Proof. Let s be the closed point of X. We have Γ(X, F ) = Fs . Hence Γ(X, −) is an exact functor. Our assertion follows. Proposition 5.7.4. Let f : X → Y be a finite morphism and let F be a sheaf on X. We have Ri f∗ F = 0 for all i ≥ 1, and H i (Y, f∗ F ) ∼ = H i (X, F ) for all i. Proof. The first assertion follows from the fact that f∗ is exact (5.3.7), and the second assertion follows from the first one and 5.6.9 (i). Let X be a scheme. For any etale sheaf F on X, define a sheaf i∗ F with respect to the Zariski topology by (i∗ F )(V ) = F (V ) for any open subset V of X. It is clear that i∗ is left exact. For every Zariski presheaf G on X, define an etale presheaf iP G by (iP G )(U ) = G (im (U → X))

for any object U in Xet . Let i∗ G = (iP G )# be the sheaf associated to iP G . We have a one-to-one correspondence Hom(G , i∗ F ) ∼ = Hom(i∗ G , F ). So i∗ is left adjoint to i∗ . Let us prove i∗ is exact. Given an exact sequence of Zariski sheaves 0 → G 0 → G → G 00 → 0, let C be the Zariski presheaf defined by C (U ) = coker(G (U ) → G 00 (U )). Then 0 → iP G 0 → iP G → iP G 00 → iP C → 0

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is an exact sequence of presheaves. So 0 → i∗ G 0 → i∗ G → i∗ G 00 → i∗ C → 0 is an exact sequence of sheaves. To prove that i∗ is exact, it suffices to show (iP C )# = 0. This follows from the fact that the Zariski sheaf associated to the Zariski presheaf C is 0. So i∗ maps injective etale sheaves to injective Zariski sheaves. For any etale sheaf F on X, we have a biregular spectral sequence p p+q E2pq = HZar (X, Rq i∗ F ) ⇒ Het (X, F ).

Proposition 5.7.5. Let X be a scheme, M a quasi-coherent OX -module, and Met the etale sheaf defined by U 7→ Γ(U, π ∗ M ) for any object π : U → q q X in Xet . We have HZar (X, M ) ∼ = Het (X, Met ) for all q. Proof. It suffices to show that Rq i∗ Met = 0 for any q ≥ 1. Note that Rq i∗ Met is the Zariski sheaf associated to the Zariski presheaf deq (U, Met ) for open subsets U of X. So it suffices to fined by U 7→ Het q show Het (U, Met ) = 0 for any q ≥ 1, any affine scheme U , and any quasi-coherent OU -module M . By 5.7.6 below, for any etale covering U = {Uα → U }α∈I with I being finite and with U and each Uα being ˇ q (U, Met ) = 0 for all q ≥ 1. So we have affine, we have H ˇ q (U, Met ) ∼ ˇ q (U, Met ) = 0 H H = lim −→ U

for all q ≥ 1. We have 1 ˇ 1 (U, Met ) Het (U, Met ) ∼ =H 1 by 5.6.3. So we have Het (U, Met ) = 0. Suppose we have shown that q Het (U, Met ) = 0 for any 1 ≤ q ≤ n. Consider the biregular spectral sequence

ˇ p (U, H q (Met )) ⇒ H p+q (U, Met ) E2pq = H et et in 5.6.2. We have p ˇ et ˇ p (U, H q (Met )), H (U, H q (Met )) = lim H −→ U

where U goes over etale coverings U = {Uα → U }α∈I of U with I being finite and with each Uα being affine. Each Uα0 ...αn is affine. So q H q (Met )(Uα0 ...αn ) = Het (Uα0 ...αn , Met ) = 0

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ˇ p (U, H q (Met )) = 0 and E pq = 0 for any p and for any 1 ≤ q ≤ n. Hence H 2 1 ≤ q ≤ n. We have an exact sequence n+1 0 → E2n+1,0 → Het (U, Met ) → E20,n+1

by [Fu (2006)] 2.2.3. Moreover, we have ˇ 0 (U, H n+1 (Met )) = H n+1 (Met )+ (U ) = 0 E20,n+1 = H et by 5.6.1. So we have n+1,0 n+1 ˇ n+1 (U, Met ) = 0. Het (U, Met ) ∼ =H = E2 et

Lemma 5.7.6. Let A → B be a faithfully flat homomorphism of rings and let M be an A-module. Then the sequence d−1

d

d

d

0 → M → M ⊗A B →0 M ⊗A B ⊗A B →1 M ⊗A B ⊗A B ⊗A B →2 · · · is exact, where dn (x ⊗ b0 ⊗ · · · ⊗ bn ) =

n+1 X i=0

(−1)i x ⊗ b0 ⊗ · · · ⊗ bi−1 ⊗ 1 ⊗ bi ⊗ · · · ⊗ bn .

Proof. Since B is faithfully flat over A, it suffices to prove that the sequence is exact after tensoring with B, that is, the sequence d−1

d

d

d

0 1 2 0→M⊗A B → M⊗A B⊗A B →M⊗ A B⊗A B⊗A B →M⊗A B⊗A B⊗A B⊗A B →···

is exact, where dn (x ⊗ b0 ⊗ · · · ⊗ bn ⊗ b) = Define

n+1 X i=0

(−1)i x ⊗ b0 ⊗ · · · ⊗ bi−1 ⊗ 1 ⊗ bi ⊗ · · · ⊗ bn ⊗ b.

Dn : M ⊗A B ⊗(n+2) → M ⊗A B ⊗(n+1) by Dn (x ⊗ b0 ⊗ · · · ⊗ bn ⊗ b) = x ⊗ b1 ⊗ · · · ⊗ bn ⊗ b0 b. Then dn−1 Dn + Dn+1 dn = id. Our assertion follows.

∗ Proposition 5.7.7. Let X be a scheme and let OX be the etale sheaf et ∗ defined by U 7→ OU (U ) for any object U in Xet . We have 1 ∗ 1 ∗ ∼ Het (X, OX )∼ (X, OX ) = Pic (X). = HZar et

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1 ∗ ∼ Proof. It is known that HZar (X, OX ) = Pic (X) ([Fu (2006)] 2.3.12). By [Fu (2006)] 2.2.3 applied to the spectral sequence p p+q ∗ ∗ E2pq = HZar (X, Rq i∗ OX ) ⇒ Het (X, OX ), et et

we have an exact sequence 1 ∗ 1 ∗ 0 ∗ 0 → HZar (X, OX ) → Het (X, OX ) → HZar (X, R1 i∗ OX ). et et

To prove our assertion, it suffices to show that the homomorphism 1 ∗ 0 ∗ Het (X, OX ) → HZar (X, R1 i∗ OX ) et et

∗ is 0. Since R1 i∗ OX is the Zariski sheaf associated to the Zariski presheaf et 1 ∗ V 7→ Het (V, OXet ) for any open subset V of X, it suffices to show that for 1 ∗ any element ξ ∈ Het (X, OX ), there exists an open covering {Vλ } of X et such that the image of ξ under the canonical homomorphism ∗ ∗ 1 1 (X, OX ) → Het (Vλ , OX ) Het et et

is 0 for every λ. The problem is local. So we may assume that X is affine. We have 1 ∗ 1 ∗ ∗ ˇ et ˇ 1 (U, OX Het (X, OX )∼ (X, OX )∼ H ), =H = lim et et et −→ et U

where U goes over the family of etale coverings of the form U = {Uα → X}α∈I such that I is finite and Uα are affine. Suppose that an element 1 ∗ ) is represented by the image of an element ξ ∈ Het (X, OX et Y Y ˇ 1 (U, O ∗ ). (ξαβ ) ∈ ker O ∗ (Uαβ ) → O ∗ (Uαβγ ) = H Xet α,β

The X-scheme U = Let

α,β,γ

`

α∈I

p : U → X,

Uα is quasi-compact and faithfully flat over X. pi : U × X U → U

pij : U ×X U ×X U → U ×X U be the projections. We have a U ×X U = Uαβ , α,β

(i = 1, 2), (1 ≤ i < j ≤ 3)

U ×X U ×X U = ∼ =

a

Uαβγ .

α,β,γ

Define an isomorphism σ : p∗1 OU → p∗2 OU so that on each Uαβ , the following diagram commutes: σ

p∗1 OU |Uαβ → p∗2 OU |Uαβ ∼ ↓∼ = ↓ = ξαβ

OUαβ → OUαβ ,

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where the vertical arrows are the canonical isomorphisms, and the bottom horizontal arrow is the multiplication by ξαβ ∈ O ∗ (Uαβ ). Since (ξαβ ) lies Q Q in the kernel of the homomorphism α,β O ∗ (Uαβ ) → α,β,γ O ∗ (Uαβγ ) in ˇ the Cech complex, we have p∗13 (σ) ∼ = p∗23 (σ) ◦ p∗12 (σ). By 1.6.1 (ii), there exist a quasi-coherent OX -module L and an isomor∼ = phism τ : p∗ L → OU such that σ ◦ p∗1 τ = p∗2 τ . By 1.6.6, L is an invertible OX -module. So there exists an open covering {Vλ } of X such that L |Vλ is isomorphic to the trivial invertible OVλ -module for each λ. One can check 1 ∗ ˇ et that the image of ξ in each H (Vλ , OX ) is 0. et Let K be a field. Choose a separable closure Ks of K. It defines a geometric point s : Spec Ks → Spec K. The galois group Gal(Ks /K) acts on Ks on the left, and hence on Spec Ks on the right. For any sheaf F on Spec K, Gal(Ks /K) acts on the stalk Fs on the left. The action is continuous with respect to the discrete topology on Fs . Indeed, we have Fs = lim F (Spec K 0 ), −→0 K

0

where K goes over the set of finite extensions of K contained in Ks , and the open subgroup Gal(Ks /K 0 ) of Gal(Ks /K) acts trivially on F (Spec K 0 ). Proposition 5.7.8. Notation as above. The functor F 7→ Fs defines an equivalence between the category of etale sheaves on Spec K and the category of Gal(Ks /K)-modules. We have H q (Spec K, F ) ∼ = H q (Gal(Ks /K), Fs ) for all q. Proof. Let K 0 be a finite separable extension of K contained in Ks . We claim that F (Spec K 0 ) ∼ = FsGal(Ks /K

0

)

for any sheaf F on Spec K. We have Fs = lim F (Spec L), −→ L

Gal(Ks /K 0 ) = lim Gal(L/K 0 ), ←− L

where L goes over the 0set of finite galois extensions of K 0 contained in Ks . Gal(Ks /K ) Given α ∈ Fs , let L be a finite galois extension of K 0 contained in Ks such that α ∈ F (Spec L). Note that Gal(Ks /K 0 ) acts on F (Spec L)

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via the finite quotient Gal(L/K 0 ) and α is invariant under the action of Gal(L/K 0 ). It follows that α lies in the kernel of the homomorphism d

F (Spec L) →0 F (Spec (L ⊗K 0 L))

ˇ in the Cech complex. Since the sequence

d

0 → F (Spec K 0 ) → F (Spec L) →0 F (Spec (L ⊗K 0 L)) Gal(K /K 0 )

s is exact, we have α ∈ F (Spec K 0 ). So we have Fs ⊂ F (Spec K 0 ). Gal(Ks /K 0 ) Gal(Ks /K 0 ) 0 0 It is clear that F (Spec K ) ⊂ Fs . So F (Spec K ) = Fs . Using this fact, one can show that for all sheaves F1 and F2 on Spec K, the canonical map

Hom(F1 , F2 ) → HomGal(Ks /K) (F1s , F2s ) is bijective. Let M be a Gal(Ks /K)-module. For any finite extension K 0 of K 0 contained in Ks , define F (Spec K 0 ) = M Gal(Ks /K ) . Every etale K-scheme is isomorphic to a disjoint union of K-schemes of the form Spec K 0 . We define a Y F ( Spec Ki ) = F (Spec Ki ) i

i

for all finite extensions Ki of K contained in Ks . Let K1 and K2 be two finite extensions of K contained in Ks , and let f : Spec K1 → Spec K2 be a K-morphism. Then f is induced by a K-homomorphism σ : K2 → K1 . We can extend σ to an element σ ¯ : Ks → Ks in Gal(Ks /K). Define the restriction F (Spec K2 ) → F (Spec K1 ) to be the homomorphism M Gal(Ks /K2 ) → M Gal(Ks /K1 ) ,

x 7→ σ ¯ (x).

This definition is independent of the choice of σ ¯ . In this way, we get a presheaf F on Spec K. For any finite galois extension L of K 0 contained in Ks , the sequence d

0 → F (Spec K 0 ) → F (Spec L) →0 F (Spec (L ⊗K 0 L)) is exact by our construction. So we have ˇ 0 ({Spec L → Spec K 0 }, F ). F (Spec K 0 ) ∼ =H

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Any etale covering U of Spec K 0 has a refinement of the form {Spec L → Spec K 0 }, and the above isomorphism coincides with the composite ˇ 0 (U, F ) → H ˇ 0 ({Spec L → Spec K 0 }, F ). F (Spec K 0 ) → H

So the homomorphism ˇ 0 (U, F ) F (Spec K 0 ) → H is injective, that is, F has the property (+) in 5.2.3. By the claim in the proof of 5.2.3 (ii), the homomorphism ˇ 0 (U, F ) → H ˇ 0 ({Spec L → Spec K 0 }, F ) H is injective. This implies that the homomorphism ˇ 0 (U, F ) F (Spec K 0 ) → H

is bijective. So F is a sheaf. We have Fs ∼ = M.

The following corollary also follows from 4.5.4. Corollary 5.7.9 (Hilbert 90). We have H 1 (Gal(Ks /K), Ks∗ ) = 0 for any field K. Proof.

Use 5.7.7, 5.7.8, and the fact that Pic(Spec K) = 0.

Let X be a scheme and let G be a sheaf of groups (not necessarily commutative) on X. Given an etale covering U = {Uα → X}α∈I of X, an Q element (gα0 α1 ) ∈ α0 ,α1 ∈I G (Uα0 α1 ) is called a 1-cocycle if gα1 α2 |Uα0 α1 α2 · gα0 α1 |Uα0 α1 α2 = gα0 α2 |Uα0 α1 α2

for any αj ∈ I (j = 0, 1, 2). Two 1-cocycles (gα0 α1 ) and (gα0 0 α1 ) are called Q equivalent if there exists (hα0 ) ∈ α0 ∈I G (Uα0 ) such that gα0 0 α1 = hα1 |Uα0 α1 · gα0 α1 · h−1 α0 |Uα0 α1 .

One checks this defines an equivalence relation on the set of 1-cocyles. We ˇ 1 (U, G ) to be the set of equivalent classes of 1-cocyles. In general, define H 1 ˇ H (U, G ) is not a group. But it has a distinguished element corresponding to the equivalent class containing the 1-cocycle (gα0 α1 ) defined by gα0 α1 = 1 for all αj ∈ I (j = 0, 1). We denote this element also by 1. Define ˇ 1 (X, G ) = lim H ˇ 1 (U, G ). H −→ U∈JX

ˇ 1 (X, G ) is (See the discussion after 5.2.1 for the definition of JX ). Again H not necessarily a group. It has a distinguished element 1.

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Let (A, a) and (A0 , a0 ) be two sets with a distinguished element. A morphism φ : (A, a) → (A0 , a0 ) between them is a map φ : A → A0 such that φ(a) = a0 . We define its kernel ker φ to be the subset φ−1 (a0 ) of A. A sequence ψ

φ

(A00 , a00 ) → (A, a) → (A0 , a0 ) is called exact if ker φ = im ψ. Proposition 5.7.10. Let 1 → G1 → G2 → G3 → 1 be an exact sequence of sheaves of groups (not necessarily commutative) on a scheme X. Then we have an exact sequence of distinguished sets 0 → Γ(X, G1 ) → Γ(X, G2 ) → Γ(X, G3 )

δ0 ˇ 1 ˇ 1 (X, G2 ) → H ˇ 1 (X, G3 ). → H (X, G1 ) → H

Suppose for any U ∈ ob Xet , G1 (U ) is mapped to the center of G2 (U ). Then G1 is a sheaf of abelian groups and we have an exact sequence of distinguished sets 0 → Γ(X, G1 ) → Γ(X, G2 ) → Γ(X, G3 ) δ

δ

0 ˇ1 1 ˇ 1 (X, G2 ) → H ˇ 1 (X, G3 ) → → H (X, G1 ) → H H 2 (X, G1 ).

Proof.

We define maps of distinguished sets ˇ 1 (X, G1 ), δ1 : H ˇ 1 (X, G3 ) → H ˇ 2 (X, G1 ) δ0 : Γ(X, G3 ) → H

and leave it for the reader to verify that the above sequences are exact. Given a section s ∈ Γ(X, G3 ), there exist an etale covering {Ui → X}i∈I and sections ti ∈ G2 (Ui ) lifting s|Ui . The sections tj |Uij · t−1 i |Uij in G2 (Uij ) are liftings of 1. So they can be lifted to sections tij ∈ G1 (Uij ). Note that ˇ 1 (X, G1 ). One can (tij ) is a 1-cocycle. We define δ0 (s) to be its image in H check δ0 is well-defined. Regard each G1 (U ) as a subgroup of G2 (U ), and suppose that it lies in ˇ 1 (X, G3 ), let U = {Ui → X}i∈I be an the center. Given an element s ∈ H etale covering of X such that s is represented by a 1-cocycle Y Y (sij ) ∈ ker G3 (Uij ) → G3 (Uijk ) . i,j∈I

i,j,k∈I

Replacing the etale covering by a refinement, we may assume that sij can be lifted to sections tij ∈ G2 (Uij ). Since sjk |Uijk · sij |Uijk = sik |Uijk ,

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there exist tijk ∈ G1 (Uijk ) such that We claim that

tjk |Uijk · tij |Uijk · t−1 ik |Uijk = tijk .

−1 tjkl |Uijkl · t−1 ikl |Uijkl · tijl |Uijkl · tijk |Uijkl = 1.

For convenience, we omit mentioning the restriction to Uijkl . We have −1 −1 −1 tjkl t−1 ikl tijl tijk = tikl tjkl tijl tijk −1 −1 −1 −1 = (til t−1 ik tkl )(tkl tjk tjl )(tjl tij til )tijk −1 −1 = (til t−1 ik )(tjk tij til )tijk −1 −1 = (til t−1 ik )tijk (tjk tij til ) −1 −1 −1 = (til t−1 ik )(tik tij tjk )(tjk tij til )

= 1. Here we use the fact that t−1 ijk lies in the center. This proves our claim. So ˇ 2 (U, G1 ) and hence an element in H ˇ 2 (X, G1 ). (tijk ) defines an element in H 2 2 ˇ (X, G1 ) ,→ H (X, G2 ). We define By 5.6.3, we have a monomorphism H δ1 (s) to be the image of (tijk ) in H 2 (X, G2 ). Let K be a field. Define an etale sheaf of groups GL(n, OSpec K,et ) on Spec K by setting GL(n, OSpec K,et )(Spec L) = GL(n, L) for any finite separable extension L of K. We have a monomorphism ∗ OSpec K,et → GL(n, OSpec K,et )

∗ which maps each section α ∈ L∗ = OSpec K,et (Spec L) to the diagonal matrix αI in GL(n, L). Let PGL(n, OSpec K,et ) be the cokernel of this monomorphism. We have a short exact sequence ∗ 1 → OSpec K,et → GL(n, OSpec K,et ) → PGL(n, OSpec K,et ) → 0.

By 5.7.10, we have a canonical map ˇ 1 (Spec K, PGL(n, OSpec K,et )) → H 2 (Spec K, O ∗ δ1 : H

Spec K,et ).

Lemma 5.7.11. Let K 0 /K be a finite separable extension of degree n, f : Spec K 0 → Spec K the corresponding morphism of schemes, and α an element in the kernel of the homomorphism ∗ 2 0 ∗ ∗ H 2 (Spec K, OSpec K,et ) → H (Spec K , f OSpec K,et ).

Then α lies in the image of the map ˇ 1 (Spec K, PGL(n, OSpec K,et )) → H 2 (Spec K, O ∗ δ1 : H

Spec K,et ).

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Proof. Note that f is a finite etale morphism. Fix a basis {e1 , . . . , en } of K 0 over K. Define a morphism ∗ f∗ f ∗ OSpec K,et → GL(n, OSpec K,et )

by mapping each section ∗ 0 ∗ s ∈ (f∗ f ∗ OSpec K,et )(Spec L) = (K ⊗K L)

to the section (aij ) ∈ GL(n, L) = GL(n, OSpec K,et )(Spec L) defined by s · (ei ⊗ 1) =

n X j=1

ej ⊗ aij .

We have a commutative diagram ∗ ∗ ∗ ∗ ∗ ∗ 0 → OSpec K,et → f∗ f OSpec K,et → f∗ f OSpec K,et /OSpec K,et → 0 k ↓ ↓ ∗ 0 → OSpec → GL(n, O ) → PGL(n, O → 0. Spec K,et Spec K,et ) K,et

It induces a commutative diagram ∗ 2 ∗ 2 ˇ 1 (Spec K, f∗ f ∗ O ∗ H Spec K,et /OSpec K,et ) → H (Spec K, OSpec K,et ) → H (Spec K, ∗ ↓ k f∗ f ∗ OSpec K,et )

ˇ 1 (Spec K, PGL(n, OSpec K,et )) H

δ

1 ∗ → H 2 (Spec K, OSpec K,et ).

Our assertion follows by a diagram chasing.

Let K be a field and A a finite dimensional algebra (not necessarily commutative) such that K is contained in the center of A. A is called a division algebra if any nonzero element in A is invertible. The opposite algebra A◦ of A has the same underlying addition group structure as A and its multiplication is defined by (a1 , a2 ) 7→ a2 a1 . We require that for any left A-module M , multiplication by 1 is the identity. A left A-module M is called simple if it is nonzero and it has no nontrivial submodule. M is called semisimple if it is a direct sum of simple modules. A is called simple if A has no nontrivial two-sided ideals. A is called a central K-algebra if K is the center of A. Lemma 5.7.12. Notation as above. Suppose M is semisimple. Then for any submodule N of M , there exists a submodule N 0 of M such that M = N ⊕ N 0.

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L Proof. Suppose M = i∈I Mi for some simple submodules Mi of M . We have either N ∩ Mi = Mi or N ∩ Mi = 0. If N ∩ Mi = Mi for all i, then M = N ⊕ 0. Otherwise, we can use Zorn’s lemma to find a nonempty subset J of I which is maximal with respect to the propL L L erty N ∩ M = 0. We claim that M = N M j j . j∈J j∈J LL It suffices to show Mi ⊂ N for all i ∈ I. If i ∈ J, j∈J Mj LL this is obvious. Suppose i 6∈ J but Mi 6⊂ N Since j∈J Mj . T LL Mi is simple, we have Mi N = 0. This implies that j∈J Mj LL N ∩ Mi = 0 which contradicts the maximality of J. j∈J Mj

Lemma 5.7.13 (Jacobson density theorem). Notation as above. Suppose M is a semisimple A-module. Let D = EndA (M ). Then M is a Dmodule. For any D-linear homomorphism f : M → M and any finitely many elements x1 , . . . , xn ∈ M , there exists a ∈ A such that f (xi ) = axi for all i. Proof. First consider the case n = 1. By 5.7.12, there exists a submodule N of M such that M = Ax1 ⊕ N . Let π : M → Ax1 be the projection. Then π ∈ D. Since f is D-linear, we have f (x1 ) = f (π · x1 ) = π · f (x1 ) = π(f (x1 )) ∈ Ax1 .

So f (x1 ) = ax1 for some a ∈ A. In general, consider the homomorphism f n : M n → M n,

(y1 , . . . , yn ) 7→ (f (y1 ), . . . , f (yn )).

Let D0 = EndA (M n ). One can check f n is D0 -linear. By the n = 1 case treated above, there exists a ∈ A such that f n (x1 , . . . , xn ) = a(x1 , . . . , xn ). Then f (xi ) = axi for all i. Lemma 5.7.14. Let K be a field and let A be a finite dimensional Kalgebra (not necessarily commutative) such that K is contained in the center of A. (i) Let M be a simple left A-module such that AnnA (M ) = 0. Then D = EndA (M ) is a finite dimensional division K-algebra and A is isomorphic KM to the algebra Mn (D◦ ) of n× n matrices with entries in D◦ for n = dim dimK D . (ii) A is a simple K-algebra if and only if A is K-isomorphic to some matrix algebra Mn (D) for some finite dimensional division K-algebra D. The integer n is uniquely determined, and D is uniquely determined up to isomorphism. D and A have the same center.

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(iii) Suppose A is a central K-algebra. Consider the K-linear map F : A ⊗K A◦ → EndK (A) defined by F (a1 ⊗ a2 )(x) = a1 xa2 . Then A is simple if and only if F is an isomorphism. (iv) Let L be a field containing K. Then K is the center of A if and only if L is the center of A ⊗K L. A is a central simple K-algebra if and only if A ⊗K L is a central simple L-algebra. (v) Suppose A is a central simple K-algebra. Let A∗ be the group of units of A and let AutK (A) be the group of K-algebra automorphisms of A. Then the homomorphism A∗ → AutK (A) defined by mapping each a ∈ A∗ to the automorphism x 7→ axa−1 induces an isomorphism A∗ /K ∗ ∼ = AutK (A). (vi) Suppose K is separably closed. Then any finite dimensional central division K-algebra is isomorphic to K. (vii) Suppose K is quasi-algebraically closed. Then any finite dimensional central K-division algebra is isomorphic to K. Proof. (i) Since M is simple, for any nonzero element x1 ∈ M , we have M = Ax1 . In particular, M is finite dimensional over K. Let f ∈ D = EndA (M ) be a nonzero element. We have ker f 6= M . Since M is simple, we must have ker f = 0. As M is finite dimensional, f must be an isomorphism. So D is a finite dimensional division K-algebra. Consider the homomorphism A → EndD (M ) that maps each a ∈ A to the endomorphism of M defined by the multiplication by a. By 5.7.13, this homomorphism is surjective. Since AnnA (M ) = 0, it is injective. So A ∼ = EndD (M ). Let n = dimD M = dimK M . Then End (M ) is isomorphic to the matrix algebra Mn (D◦ ). D dimK D (ii) Suppose A is a simple algebra. Note that simple A-modules exist. In fact, any nonzero left ideal of A with minimal dimension is a simple A-module. For any simple A-module M , AnnA (M ) is a two-sided ideal of A and hence is 0. By (i), A is isomorphic to a matrix algebra Mn (D) for some finite dimensional division K-algebra D. Let I be a nonzero two-sided ideal of Mn (D). For any 1 ≤ i, j ≤ n, denote by Eij the matrix whose (i, j)-entry is 1 and whose other entries

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are 0. Let (aij ) be a nonzero element in I and suppose ai0 j0 6= 0. Then we have Eij = a−1 i0 j0 Eii0 · (aij ) · Ej0 j ∈ I

for any 1 ≤ i, j ≤ n. It follows that I = Mn (D) and hence Mn (D) is a simple algebra. For each 1 ≤ j ≤ n, let Jj be the left ideal of Mn (D) consisting of matrices whose nonzero entries are on the j-th column. Note that Jj are isomorphic to each other as left Mn (D)-modules since they are all isomorphic to the space of column vectors with n-entries. Let (aij ) be an arbitrary nonzero element in J1 . Suppose ai0 1 6= 0. Then we have Ei1 = a−1 i0 1 Eii0 · (aij ) ∈ J1

for any i. It follows that J1 is generated by any nonzero element in it, and hence is a simple left Mn (D)-module. Any simple left Mn (D)-module M is isomorphic to J1 . Indeed, let x be a nonzero element in M . For each 1 ≤ j ≤ n, Ann(x) ∩ Jj is a left-submodule of Jj . We have either Ann(x) ∩ Jj = 0 or Ann(x) ∩ Jj = Jj . If Ann(x) ∩ Jj = Jj for all j, then Mn (D) = J1 + · · · + Jn ⊂ Ann(x).

This is impossible since x 6= 0. So we have Ann(x) ∩ Jj = 0 for some j. Let us prove that M is then isomorphic to Jj , and hence isomorphic to J1 . Consider the homomorphism φ : Jj → M,

a 7→ ax.

We have ker φ = 0. The image of φ is a nonzero submodule of M . Since M is simple, we have im φ = M . So φ is an isomorphism. We claim that D◦ ∼ = EndMn (D) (M ) for any simple left Mn (D)-module M . So D is uniquely determined up to isomorphism by the algebra Mn (D). We can take M = J1 . For any matrix X in J1 , we have X = XE11 . For any f ∈ EndMn (D) (J1 ), we have f (X) = f (XE11 ) = Xf (E11 ) = X(E11 f (E11 )).

Note that the only nonzero entry of E11 f (E11 ) is the (1, 1)-entry. We get an isomorphism between EndMn (D) (J1 ) and D◦ by mapping f to the (1, 1)entry of E11 f (E11 ). We leave it for the reader to prove that Mn (D) and D have the same center. (iii) If A has a nontrivial two-sided ideal I, then for any element α ∈ A ⊗K A◦ , we have F (α)(I) ⊂ I. But there are K-endomorphisms of A not mapping I to I. So F cannot be an isomorphism.

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Suppose A is simple. Let M be the left (A ⊗K A◦ )-module whose underlying addition group is A and whose scalar multiplication is given by α · x = F (α)(x) for any α ∈ A⊗K A◦ and x ∈ M . Note that submodules of M are two-sided ideals of A. So M is a simple left (A ⊗K A◦ )-module. Let φ : M → M be an (A ⊗K A◦ )-module homomorphism, that is, φ : A → A is a K-linear map such that φ(axb) = aφ(x)b for any a, b, x ∈ A. Taking x = b = 1, we get φ(a) = aφ(1). Taking a = x = 1, we get φ(b) = φ(1)b. So we have aφ(1) = φ(1)a, and hence φ(1) lies in the center K of A. We can thus identify EndA⊗K A◦ (M ) with K. So M is a simple left module with trivial annihilator over (A ⊗K A◦ )/AnnA⊗K A◦ (M ), and we have End(A⊗K A◦ )/AnnA⊗

◦ (M) KA

(M ) ∼ = EndA⊗K A◦ (M ) ∼ = K.

By (i), we have (A ⊗K A◦ )/AnnA⊗K A◦ (M ) ∼ = Mn (K) for n = dimK M = dimK A. So dimK ((A ⊗K A◦ )/AnnA⊗K A◦ (M )) = n2 . But dimK (A ⊗K A◦ ) = n2 . It follows that AnnA⊗K A◦ (M ) = 0, that is, ker F = 0. As A ⊗K A◦ and EndK (A) have the same dimension, F is an isomorphism. (iv) Let C be the center of A. Fix a basis {a1 , . . . , an } of A over K. Consider the K-linear map φ : A → An ,

φ(a) = (aa1 − a1 a, . . . , aan − an a).

Then C = ker φ and hence C ⊗K L ∼ = ker (φ ⊗ idL ). As ker (φ ⊗ idL ) is the center of A ⊗K L, this shows that C ⊗K L can be identified with the center of A ⊗K L. So K is the center of A if and only if L is the center of A ⊗K L. Using (iii), one can show A is a central simple K-algebra if and only if A ⊗K L is a central simple L-algebra.

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(v) Fix a basis {a1 , . . . , an } of A over K. Let σ be an element in AutK (A). By (iii), σ is of the form x 7→

n X

ai xbi

i=1

for some bi ∈ A. From σ(xy) = σ(x)σ(y), we get n X i=1

ai x(ybi − bi σ(y)) = 0.

By (iii) again, this implies that ybi − bi σ(y) = 0 for all i and all y ∈ A. So each bi A is a two-sided ideal of A. Hence bi A = 0 or bi A = A, that is, bi = 0 or bi is invertible. If some bi is invertible, then we have σ(y) = b−1 i ybi . If no bi is invertible, all bi vanish and σ = 0. This is impossible since σ is an automorphism. So σ is of the form x 7→ axa−1 for some unit a in A. If a is a unit in A such that axa−1 = x for all x ∈ A, then a lies in the center of A. Hence a ∈ K ∗ . We thus have A∗ /K ∗ ∼ = AutK (A). (vi) Let D be a finite dimensional central division K-algebra. Suppose K is algebraically closed. For any α ∈ D, the subalgebra K[α] of D generated by K and α is commutative and finite dimensional over K. It is a field. Since K is algebraically closed, we have K[α] = K. So α ∈ K. We thus have K = D. By (ii), any finite dimensional central simple K-algebra is isomorphic to the matrix algebra Mn (K) for some n. Now suppose K is separably closed. For any element α ∈ D, K[α] is field finite and purely inseparable over K. Let p = char K. Then the degree m of α over K is pm for some positive integer m and αp ∈ K. Let q be the largest power of p dividing dimK D. Then pm |q and hence αq ∈ K. Fix a decomposition D = K ⊕ E as K-vector spaces. Let K be an algebraic closure of K. This decomposition induces a decomposition D ⊗K K = K ⊕ (E ⊗K K). Let π : D ⊗K K → E ⊗K K be the projection. The map φ : D ⊗K K → E ⊗K K, x 7→ π(xq )

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is a polynomial map defined over K, that is, if we fix bases for D and E over K and express this map in terms of coordinates, then we get a polynomial map defined by polynomials with coefficients in K. By the above discussion, we have φ(x) = 0 for any x ∈ D. As K is necessarily an infinite field, this implies that φ(x) = 0 for any x ∈ D ⊗K K, that is, xq ∈ K for any x ∈ D ⊗K K. On the other hand, by the discussion for the algebraically closed field case, we have D ⊗K K ∼ = Mn (K) for some n ≥ 1. q does not lie in the image of K in Mn (K) For the matrix E11 in Mn (K), E11 if n ≥ 2. So we have n = 1 and hence D = K. (vii) We first introduce the reduced norm. Let K be a field, let Ks be a separable closure of K, and let A be a finite dimensional central simple K-algebra. By (ii), (iv) and (vi), we have an isomorphism of Ks -algebras φ : A ⊗K Ks ∼ = Mn (Ks ) for some integer n. In particular, dimK A is the square of an integer. For any a ∈ A, we defined the reduced norm Nrd(a) of a to be the determinant of φ(a ⊗ 1). By (v), if φ0 : A ⊗K Ks ∼ = Mn (Ks )

is another isomorphism of Ks -algebras, then there exists an invertible matrix P such that φ0 = P −1 φP. It follows that det (φ(a ⊗ 1)) = det (φ0 (a ⊗ 1)).

So Nrd(a) is independent of the choice of the isomorphism φ. We will prove shortly that Nrd(a) ∈ K for any a ∈ A. As Nrd : A → K is the restriction to A of the map det A ⊗K Ks ∼ = Mn (Ks ) → Ks ,

√ it is given by a homogeneous polynomial of degree dimK A. Let us prove that Nrd takes values in K. There exists a finite galois extension K 0 of K contained in Ks such that we have an isomorphism of K 0 -algebras. φ : A ⊗K K 0 ∼ = Mn (K 0 ). For any σ ∈ Gal(K 0 /K), let id ⊗ σ and Mn (σ) be the automorphisms induced by σ on A ⊗K K 0 and Mn (K 0 ) respectively. Then Mn (σ) ◦ φ ◦ (id ⊗ σ −1 ) : A ⊗K K 0 → Mn (K 0 )

is also an isomorphism of K 0 -algebras. By (v), there is an invertible matrix P in Mn (K 0 ) such that Mn (σ) ◦ φ ◦ (id ⊗ σ −1 ) = P φP −1 .

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For any a ∈ A, we have

det(Mn (σ) ◦ φ ◦ (id ⊗ σ −1 )(a ⊗ 1)) = σ(det(φ(a ⊗ 1))).

It follows that σ(det(φ(a ⊗ 1))) = det(P φ(a ⊗ 1)P −1 ) = det(φ(a ⊗ 1)), that is, σ(Nrd(a)) = Nrd(a). So we have Nrd(a) ∈ K. Now let us prove (vii). Suppose K is quasi-algebraically closed. Let D √ be a finite dimensional central division K-algebra, and let n = dimK D. The reduced norm Nrd : D → K is given by a homogeneous polynomial of degree n with n2 variables. First consider the case where K is infinite. Then the coefficients of the polynomial expressing the map Nrd lie in K. For any x ∈ D − {0}, we have Nrd(x) · Nrd(x−1 ) = 1.

So Nrd(x) has no nontrivial zero in D. Since K is quasi-algebraically closed, the number of variables of the polynomial expressing Nrd does not exceed the degree, that is, n2 ≤ n. So n = 1 and hence D = K. In the case where K is finite, D is a finite division algebra. A theorem of Wedderburn says that D must be a field. Since K is the center of D, we have K = D. The following is 4.5.6. Proposition 5.7.15. Suppose that K is a quasi-algebraically closed field. We have H 2 (Gal(Ks /K), Ks∗ ) = 0. Proof.

By 5.7.8, we need to show ∗ H 2 (Spec K, OSpec K,et ) = 0.

∗ · Let α ∈ H 2 (Spec K, OSpec K,et ). Choose an injective resolution I of ∗ OSpec K,et and suppose that α is given by the cohomology class of a section s ∈ Γ(Spec K, I 2 ) whose image in Γ(Spec K, I 3 ) is 0. There exists a finite separable extension K 0 of K such that the restriction of s to Spec K 0 can be lifted to a section in Γ(Spec K 0 , I 1 ). The image of α under the canonical homomorphism ∗ 2 0 ∗ H 2 (Spec K, OSpec K,et ) → H (Spec K , OSpec K 0 ,et )

is then 0. By 5.7.11, α lies in the image of the canonical homomorphism ˇ 1 (Spec K, PGL(n, OSpec K,et )) → H 2 (Spec K, O ∗ H ), Spec K,et

0

where n = [K : K]. To prove our assertion, it suffices to show ˇ 1 (Spec K, PGL(n, OSpec K,et )) = 0. H

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Let L be a finite separable extension of K. Then {Spec L → Spec K} is an etale covering of Spec K, and any etale covering of Spec K has a refinement of this form. Let σ ∈ PGL(n, L ⊗K L) be a 1-cocyle for this etale covering and the sheaf PGL(n, OSpec K,et ). Consider the sheaf of matrix algebra Mn (OSpec L ) over OSpec L . Let pi : Spec (L ⊗K L) → Spec L

(i = 1, 2),

pij : Spec (L ⊗K ⊗K L) → Spec (L ⊗K L) (1 ≤ i < j ≤ 3) be the projections. Then the conjugation by σ ∈ PGL(n, L ⊗K L) defines an automorphism σ : p∗1 Mn (OSpec L ) → p∗2 Mn (OSpec L ) such that p∗13 (σ) = p∗23 (σ) ◦ p∗12 (σ). So σ defines a descent datum for the matrix algebra Mn (L) over L. By 1.6.1, 5.7.14 (iv) and (v), this matrix algebra can be descended down to a central simple K-algebra A. By 5.7.14 (ii) and (vii), A is isomorphic to Mn (K). So the descent datum is isomorphic to the trivial one and the 1-cocycle is equivalent to the distinguished 1-cocycle. Hence ˇ 1 (Spec K, PGL(n, OSpec K,et )) = 0. H Let X be a scheme. An X-scheme G is called a group scheme if for any X-scheme Y , the set HomX (Y, G) has a group structure, and for any X-morphism of X-schemes f : Y1 → Y2 , the map HomX (Y2 , G) → HomX (Y1 , G),

g 7→ gf

is a homomorphism of groups. A right action of G on an X-scheme P is an X-morphism P ×X G → P such that for any X-scheme Y , this X-morphism induces a group action HomX (Y, P ) × HomX (Y, G) → HomX (Y, P ). We often denote this X-morphism by P ×X G → P,

(p, g) 7→ pg.

We say that G is an etale (resp. flat, resp. smooth) group scheme if the structure morphism G → X is etale (resp. flat, resp. smooth). Proposition 5.7.16. Let G be an etale group scheme over X, let P be an X-scheme on which G acts on the right, and let F : P ×X G → P ×X P,

(p, g) 7→ (p, pg)

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be the morphism so that π1 F is the projection P ×X G → P,

(p, g) 7→ p

to the first factor, and π2 F is the group action P ×X G → P,

(p, g) 7→ pg,

where π1 , π2 : P ×X P → P are the two projections. The following conditions are equivalent: (i) The structure morphism P → X is surjective and etale, and the morphism F : P ×X G → P ×X P is an isomorphism. (ii) There exists an etale covering {Ui → X}i∈I such that for any i ∈ I, there exists a Ui -isomorphism Ui ×X G ∼ = Ui ×X P which is compatible with the canonical right actions of Ui ×X G on itself and on Ui ×X P . (iii) The structure morphism P → X is surjective and etale, and for any X-scheme Y , the action of the group HomX (Y, G) on HomX (Y, P ) is transitive, and the stabilizer of any element in HomX (Y, P ) is trivial. Proof. (i)⇒(ii) Use the etale covering {P → X}. (ii)⇒(i) The problem is local and we may assume that X is affine, I ` is finite, and each Ui → X has finite presentation. Let U = i∈i Ui . Then U → X is faithfully flat and quasi-compact. The unit in the group HomX (U, G) defines a section for the projection U ×X G → U . In particular, this projection is surjective. Since G is etale over X, this projection is etale. But U ×X G ∼ = U ×X P as U -schemes. So U ×X P → U is surjective and etale. By 1.7.2 and 2.5.10, P → X is surjective and etale. Its clear that G ×X G → G ×X G,

(g 0 , g) 7→ (g 0 , g 0 g)

is an isomorphism. Since U ×X G ∼ = U ×X P as U -schemes with group actions, the base change of the morphism P ×X G → P ×X P,

(p, g) 7→ (p, pg)

by U → X is an isomorphism. So this morphism is an isomorphism by 1.8.4.

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(i)⇔(iii) The condition that the action of the group HomX (Y, G) on HomX (Y, P ) is transitive, and that the stabilizer of any element in HomX (Y, P ) is trivial is equivalent to the condition that the map HomX (Y, P ) × HomX (Y, G) → HomX (Y, P ) × HomX (Y, P )

induced by the morphism

P ×X G → P ×X P,

is an isomorphism.

(p, g) 7→ (p, pg)

When P satisfies the equivalent conditions of 5.7.16, we say that P is a G-torsor. We say that P is a trivial G-torsor if P is X-isomorphic to G. Proposition 5.7.17. Let G be an etale group scheme over X. A G-torsor P is trivial if and only if the structure morphism P → X has a section. Proof. The unit in the group HomX (X, G) defines a section for the structure morphism G → X. Suppose P → X has a section s. Consider the composite s×id

G∼ = X ×X G → P ×X G → P,

where the second morphism is the group action. Using condition (iii) in 5.7.16, one verifies that this morphism induces a bijection HomX (Y, G) → HomX (Y, P )

for any X-scheme Y . It follows that it is an isomorphism.

e be the sheaf on X Let G be an etale group scheme over X, and let G e on represented by G. Let F be a sheaf of sets on X. A right action of G F is a morphism e→F F ×G such that for any object U in Xet , the map e ) → F (U ) F (U ) × G(U

e ) on F (U ). We say that F is a G-torsor e defines a right action of G(U if there exists an etale covering {Ui → X} of X such that for each i, we have e Ui on itself e Ui ∼ an isomorphism G| = F |Ui compatible with the actions of G| e and on F |Ui . Note that if P is a G-torsor, then the sheaf P represented e by P is a G-torsor. Proposition 5.7.18. Suppose G is a separated etale group scheme with finite presentation over a scheme X. Then the functor P 7→ Pe from e the category of G-torsors to the category of G-torsors is an equivalence of categories.

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Proof.

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Let P1 and P2 be G-torsors. Define a map

T : Hom(Pe1 , Pe2 ) → HomX (P1 , P2 ) as follows. Let φ : Pe1 → Pe2 be a morphism of sheaves. idP1 is a section of Pe1 over P1 ∈ ob Xet . We define the X-morphism T (φ) : P1 → P2 to be the image of idP1 under the map φ(P1 ) : Pe1 (P1 ) → Pe2 (P1 ). One checks that T is the inverse of the canonical map HomX (P1 , P2 ) → Hom(Pe1 , Pe2 ).

Hence the functor P 7→ Pe from the category of G-torsors to the category e of G-torsors is fully faithful. e Let F be a G-torsor. Let us prove there exists a G-torsor P such that ∼ e F = P . Let {Vi }i∈I be an affine open covering of X. If we can find G∼ = torsors Pi on Vi such that we have isomorphisms φi : F |Vi → Pei , then by the discussion above, the isomorphisms φj ◦ φ−1 define isomorphisms i ∼ φij : Pi |V ∩V = Pj |V ∩V i

j

i

j

such that φjk φij ∼ = φik on Vi ∩ Vj ∩ Vk . We can glue Pi together to get a G-torsor P on X such that F ∼ = Pe. We are thus reduced to the case where X is affine. By 1.10.12 and 2.3.7, G is quasi-affine over X. We can find an etale covering {Uj → X}j∈J such that J is finite, each Uj → X has finite presentation and F |Uj is e Uj -torsor. Let U = ` isomorphic to the trivial G| j∈J Uj . Then U → X is quasi-compact and faithfully flat, and F |U is represented by the trivial G|U -torsor P 0 . The descent datum on F |U defining the sheaf F defines a descent datum for the G-torsor P 0 . By 1.8.11, P 0 can be descended down to a G-torsor P on X. We then have F ∼ = Pe. Proposition 5.7.19. Let G be a separated etale group scheme (not necessarily commutative) with finite presentation over X. Then the set of ˇ 1 (X, G). e isomorphic classes of G-torsors is isomorphic to H

e Proof. By 5.7.18, it suffices to show that the isomorphic classes of G1 ˇ (X, G). e Let F be a G-torsor e torsors is isomorphic to H and let {Ui → X}i∈i be an etale covering such that we have isomorphisms ∼ =

e Ui φi : F |Ui → G|

e Ui -torsors. For each pair (i, j), φj ◦ φ−1 |Ui ×X Uj is an isomorphism of of G| i e Ui ×X Uj -torsor. So there exists a section sij ∈ G(U e i ×X Uj ) the trivial G|

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such that φj ◦ φ−1 i |Ui ×X Uj is induced by left multiplication by sij . One can verify that (sij ) is a 1-cocycle. We define a map from the set of isomorphic e ˇ 1 (X, G) e by assigning the image of the 1classes of G-torsors to the set H 1 ˇ e cocyle (sij ) in H (X, G) to F . One checks that this map is bijective.

Let G be a finite group (not necessarily commutative). For any scheme ` X, let GX = g∈G Xg , where each Xg is a copy of X. For any X-scheme Y , we have a HomX (Y, GX ) = HomX (Y, Xg ). g∈G

` But HomX (Y, Xg ) has only one element. So g∈G HomX (Y, Xg ) can be identified with G and GX is a group scheme over X. Assume that X is a connected noetherian scheme and γ : s → X is a geometric point of X. For every X-scheme X 0 , let X 0 (γ) be the set of geometric points of X 0 lying above γ. By 3.2.12, the functor X 0 7→ X 0 (γ) is an equivalence from the category of etale covering spaces of X to the category of finite sets on which π1 (X, γ) acts continuously on the left. Let P be a GX -torsor. Then P is an etale covering spaces of X. So π1 (X, γ) acts continuously on P (γ) on the left. On the other hand, G acts on P (γ) on the right. Note that these two actions on P (γ) commute with each other, and the map P (γ) × G → P (γ) × P (γ),

(p, g) 7→ (p, pg)

is bijective, that is, the action of G on P (γ) is transitive and stabilizers of points in P (γ) are trivial. Conversely, let A be a finite set on which π1 (X, γ) acts continuously on the left and G acts on the right such that these two actions commute with each other, G acts transitively, and stabilizers of points in A are trivial. Then there exists an etale covering space P of X such that P (γ) is isomorphic to A as π1 (X, γ)-sets. The action of G on A induces a homomorphism G → Aut(P/X)◦ , and hence a morphism P ×X GX → P. Since the map A × G → A × A,

(a, g) 7→ (a, ag)

is bijective, the morphism P ×X GX → P ×X P, is an isomorphism. So P is a GX -torsor.

(p, g) 7→ (p, pg)

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The set of isomorphic classes of GX -torsors is thus isomorphic to the isomorphic classes of finite sets on which π1 (X, γ) acts continuously on the left and G acts on the right, such that these two actions commute with each other, G acts transitively, and stabilizers of points in A are trivial. Any finite set A on which G acts transitively on the right such that stabilizers of points in A are trivial is isomorphic to G with G acting on it by right multiplication. Suppose π1 (X, γ) acts on G on the left such that this action commutes with the right multiplication on G. It is completely determined by the map π1 (X, γ) → G,

σ 7→ σe.

Indeed, for any g ∈ G and σ ∈ π1 (X, γ), we have σ(g) = σ(eg) = (σe)g. This map is a continuous homomorphism. Indeed, for any σ1 , σ2 ∈ π1 (X, γ), we have (σ1 σ2 )e = σ1 (σ2 e) = (σ1 e)(σ2 e). Conversely, any continuous homomorphism from π1 (X, γ) to G defines a continuous left action of π1 (X, γ) on G which commutes with the right multiplication on G. We thus prove the following. Proposition 5.7.20. Let X be a connected noetherian scheme, γ a geometric point in X, and G a finite group. Then we have ˇ 1 (X, G eX ) ∼ H = cont.Hom(π1 (X, γ), G),

where cont.Hom(π1 (X, γ), G) is the set of continuous homomorphisms from π1 (X, γ) to G. 5.8

Constructible Sheaves

([SGA 4] IX 2.) Let X be a scheme. For any set G, the constant sheaf on X associated to G is the sheaf associated to the constant presheaf U 7→ G for any U ∈ ob Xet . We denote this sheaf also by G. For any connected object U in Xet , we have ` G(U ) = G. The sheaf G is represented by the X-scheme GX = g∈G Xg , where each Xg is a copy of X. A sheaf F is called locally constant if there exists an etale covering {Ui → X}i∈I such that F |Ui are constant sheaves.

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Proposition 5.8.1. Let X be a scheme and let F be a sheaf of sets on X. (i) Suppose that the stalks of F are finite. Then F is locally constant if and only if it is represented by an etale covering space of X. Moreover, if F is locally constant and has finite stalks, then there exists a surjective finite etale morphism π : Y → X such that π ∗ F is constant. (ii) If F is locally constant and X has finitely many irreducible components, then there exist a dense open subset V of X and a surjective finite etale morphism π : V 0 → V such that π ∗ (F |V ) is constant. Proof. (i) Let F be a locally constant sheaf with finite stalks. Let us prove e 0 . Let that there exists an etale covering space X 0 → X such that F ∼ =X {Vi }i∈I be an affine open covering of X. If we can find etale covering spaces πi : Vi0 → Vi such that we have isomorphisms φi : F |Vi ∼ = Vei0 ,

then φj ◦ φ−1 i |Vi ∩Vj induce isomorphisms ∼ =

φij : πi−1 (Vi ∩ Vj ) → πj−1 (Vi ∩ Vj ) and φjk φij ∼ = φik over Vi ∩ Vj ∩ Vk . We can glue Vi0 together to get an e 0 . We are thus reduced to etale covering space X 0 of X such that F ∼ =X the case where X is affine. We can find an etale covering {Uj → X}j∈J such that J is finite, each Uj → X has finite presentation and each F |Uj ` is a constant sheaf with finite stalks. Let U = j∈J Uj . Then U → X is quasi-compact and faithfully flat, and F |U is represented by a trivial etale covering space U 0 of U . The descent datum on F |U defining the sheaf F defines a descent datum for U 0 . By 1.8.6, U 0 can be descended down to an X-scheme X 0 . By 1.8.8 and 2.5.10, X 0 is an etale covering space of X. We e 0. then have F ∼ =X Conversely, let f : X 0 → X be an etale covering space. By 5.2.7 (iii), e 0 at a geometric point γ : s → X can be identified with the set the stalk of X X 0 (γ) of geometric points of X 0 lying above γ. So it is finite. Let us prove e 0 is locally constant, and that there exists a finite surjective etale that X e 0 is constant. We may assume that X 0 morphism π : Y → X such that π ∗ X and X are connected. It suffices to show that there exists a surjective finite etale morphism π : Y → X such that as a Y -scheme, X 0 ×X Y is a disjoint union of copies of Y . Note that im f is an open and closed subset of X and hence coincides with X. So f is surjective. The diagonal morphism ∆ : X 0 → X 0 ×X X 0 is an open and closed immersion. By induction on the

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number of elements of the set X 0 (γ) and applying the induction hypothesis to the etale covering space X 0 ×X X 0 − im ∆ → X 0

induced by the projection X 0 ×X X 0 → X 0 to the first factor, we may assume that there exists a surjective finite etale morphism Y → X 0 such that (X 0 ×X X 0 − im ∆) ×X 0 Y is a disjoint union of copies of Y . Then the composite Y → X 0 → X is a finite surjective etale morphism with the required property. (ii) Let η1 , . . . , ηm be all the generic points of X. For each ηi , let Vi be an affine open neighborhood of ηi in X such that Vi ∩ Vj = Ø for i 6= j. Choose an etale morphism πi : Vi0 → Vi of finite presentation such that πi∗ (F |Vi ) is constant. The generic fiber πi−1 (ηi ) → ηi of πi is finite, surjective, and etale. By 1.10.10 (iv) and (vi) and 2.3.7, after shrinking Vi , we may assume that πi is finite, surjective, and etale. Let π be the ` ` ` morphism πi : V 0 = Vi0 → V = Vi . Then π is a surjective finite etale morphism and π ∗ (F |V ) is constant. Let X be a noetherian scheme and A a noetherian ring. A sheaf F of sets (resp. A-modules) is called constructible if there exists a decomposition Sn X = i=1 Xi of X into finitely many locally closed subsets Xi such that each F |Xi is locally constant and the stalks of F are finite (resp. finitely generated A-modules). A sheaf of abelian groups F on X is called a torsion sheaf if F (U ) are torsion abelian groups of all U ∈ ob Xet . A sheaf of abelian groups on X is called constructible if it is constructible as a sheaf of sets. Any constructible sheaf of abelian group F on X is a torsion sheaf, and there exists a nonzero integer n such that nF = 0. Note that a sheaf of Z-modules is constructible as a sheaf of abelian groups if and only if it is a torsion sheaf and is constructible as a sheaf of Z-modules. The constant sheaf Z is a constructible sheaf of Z-modules, but not a constructible sheaf of abelian groups. Proposition 5.8.2. Let f : X 0 → X be a morphism of noetherian schemes, A a noetherian ring, and F a sheaf of sets (resp. abelian groups, resp. Amodules). If F is constructible, then so is f ∗ F . Proposition 5.8.3. Let X be a noetherian scheme, A a noetherian ring, and F a sheaf of sets (resp. abelian groups, resp. A-modules). F is constructible if and only if for any irreducible closed subset Y of X, there exists a nonempty open subset V of Y such that F |V is locally constant with finite (resp. finite, resp. finitely generated) stalks.

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Proof. Suppose F is constructible. For any irreducible closed subset Sn Y , F |Y is constructible. So there exists a decomposition Y = i=1 Yi of Y into finitely many locally closed subsets Yi such that F |Yi are locally constant. Let Yi0 be a locally closed subset containing the generic point of Y . Then Yi0 is a nonempty open subset of Y and F |Yi0 is locally constant. Conversely, suppose for any irreducible closed subset Y of X, that there exists a nonempty open subset V of Y such that F |V is locally constant with finite (resp. finite, resp. finitely generated) stalks. Let S = {Y |Y ⊂ X is nonemtpy and closed and F |Y is not constructible}. If S 6= Ø, then since X is noetherian, S has a minimal element Y . If Y is not irreducible, then Y = Y1 ∪ Y2 for two proper closed subsets Yi (i = 1, 2). By the minimality of Y , F |Yi are constructible, and hence F |Y is constructible. This contradicts the fact that Y ∈ S . So Y is irreducible. By our assumption, there exists a nonempty open subset V of Y such that F |V is constructible. By the minimality of Y , F |Y −V is constructible. It follows that F |Y is constructible. Contradiction. So S = Ø. Hence F is constructible. Proposition 5.8.4. Let X be a noetherian scheme, f : U → X an etale morphism of finite type, and A a noetherian ring. e represented by U is a constructible sheaf of sets. (i) The sheaf U (ii) The sheaf AU = f! A is a constructible sheaf of A-modules.

Proof. Let Y be an integral closed subscheme of X and let K be its function field. Then n a U ×X Spec K ∼ Spec Li = i=1

for some finite separable extensions Li of K. By 1.10.10 (iv) and 2.3.7, there exists a nonempty open subset V of Y such that U ×X V is finite e |V is represented by the V -scheme U ×X V by etale over V . Note that U e |V is locally constant with finite stalks. So by 5.2.7 (iii). By 5.8.1 (i), U e 5.8.3, U is constructible. Let f 0 : U ×X V → V be the base change of f . By 5.5.1 (ii), we have (f! A)|V ∼ = f 0 A. !

By the proof of 5.8.1 (i), there exists an etale covering {Vi → V }i∈i such that Vi are connected and U ×X Vi are disjoint union of copies of Vi . It follows that (f!0 A)|Vi are constant sheaves of free A-modules with finite rank, and hence f!0 A is locally constant. By 5.8.3, f! A is constructible.

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Let X be a noetherian scheme and A a noetherian ring. A sheaf F of sets (resp. abelian groups, resp. A-modules) is called noetherian if any ascending chain of subsheaves of sets (resp. abelian groups, resp. Amodules) F1 ⊂ F2 ⊂ · · · ⊂ F in F is stationary, that is, Fi = Fi+1 = · · · for large i. Proposition 5.8.5. Let X be a noetherian scheme and A a noetherian ring. (i) For any sheaf of A-modules F on X, the following conditions are equivalent: (a) F is a constructible sheaf of A-modules. (b) F is noetherian. (c) There exists an exact sequence of the form AV → AU → F → 0, where AU = f! A and AV = g! A for some etale morphisms f : U → X and g : V → X of finite type. (ii) For any sheaf of abelian groups F on X, the following conditions are equivalent: (a) F is a constructible sheaf of abelian groups. (b) F is a torsion sheaf and noetherian. (c) F is a constructible sheaf of Z/n-modules for some nonzero integer n. Proof. (i) (a)⇒(b) Let F1 ⊂ F2 ⊂ · · · be an ascending chain of subsheaves of F and let S = {Y | Y ⊂ X is nonempty and closed and

F1 |Y ⊂ F2 |Y ⊂ · · · is not stationary}.

If S 6= Ø, then S has a minimal element Y , and Y is necessarily irreducible. By 5.8.3, there exists a nonempty open subset V of Y such that F |V is locally constant with finitely generated stalks. Let η ∈ V be the generic point of Y and let s1¯η , . . . , sn¯η be a finite family of generators S∞ for the A-module i=1 Fi¯η . Choose i0 sufficiently large, we may assume s1¯η , . . . , sn¯η ∈ Fi0 η¯ . We may find an etale neighborhood U of η¯ in V such

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that s1¯η , . . . , sn¯η are germs at η¯ of sections s1 , . . . , sn ∈ Fi0 (U ), respectively. It follows that if x lies in the image of U in V and i ≥ i0 , then s1¯η , . . . , sn¯η lie in the image of specialization homomorphism Fi¯x → Fi¯η , and hence the specialization homomorphism is surjective for any i ≥ i0 and any x ∈ im (U → V ). Since F |V is locally constant, the specialization homomorphism Fx¯ → Fη¯ is bijective, and hence the specialization homomorphism Fi¯x → Fi¯η is injective for any i. So the specialization homomorphism Fi¯x → Fi¯η is bijective for any i ≥ i0 and any x ∈ im (U → V ). We have (Fi0 )η¯ = (Fi0 +1 )η¯ = · · · . So we have (Fi0 )x¯ = (Fi0 +1 )x¯ = · · · for any x ∈ im (U → V ). By the minimality of Y , the chain F1 ⊂ F2 ⊂ · · · is stationary on Y − im (U → V ). Hence the chain is stationary on Y . This contradicts the fact that Y ∈ S . So S = Ø and the chain is stationary. (b)⇒(c) For any etale X-scheme U , we have canonical one-to-one correspondences Hom(AU , F ) ∼ = Hom(A, F |U ) ∼ = F (U ). For any section s ∈ F (U ), let φs : AU → F be the morphism corresponding to s. It maps the section 1 in AU (U ) to s. We can find a set {Uα } of objects Uα in Xet with the property that Uα are affine and the images of Uα in X are contained in affine open subsets of X such that any object in Xet with this property is isomorphic to some Uα . Then we have an epimorphism M ⊕φs AUα → F . α,s∈F (Uα )

Since F is noetherian, we can find finitely many objects Ui (i = 1, . . . , n) in {Uα } and sections si ∈ F (Ui ) such that n M i=1

⊕φs

AUi → i F

` is an epimorphism. Let U = ni=1 Ui . We then have an epimorphism φ : AU → F . By 5.8.4, AU is constructible and hence noetherian. So ker φ is noetherian. The above argument then shows that there exists an epimorphism AV → ker φ for some etale X-scheme V of finite type. We then have an exact sequence AV → AU → F → 0.

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(c)⇒(a) By 5.8.4, AU and AV are constructible. So there exists a deSn composition X = i=1 Xi of X into finitely many locally closed subsets such that AU |Xi and AV |Xi are locally constant. Let Xi0 → Xi be surjective etale morphisms such that AU |Xi0 and AV |Xi0 are constant. Then the inverse image of F on each connected component of Xi0 is constant. So F is constructible. Sn (ii) (a)⇒(c) Let X = i=1 Xi be a decomposition of X into finitely many locally closed subsets such that F |Xi are locally constant sheaves with finite stalks. We can find a nonzero integer n such that each F |Xi is a sheaf of Z/n-modules. Then F is a constructible sheaf of Z/n-modules. (c)⇒(b) follows from (i) (a)⇒(b). (b)⇒(a) Since F is a torsion sheaf, we have [ F = ker (n : F → F ). n∈N

Since F is noetherian, there exists a nonzero integer n such that F = ker (n : F → F ). Then F is a sheaf of Z/n-modules. By (i) (b)⇒(a), F is a constructible sheaf of Z/n-modules. This implies that F is a constructible sheaf of abelian groups. Proposition 5.8.6. Let X be a noetherian scheme and A a noetherian ring. (i) Subsheaves and quotient sheaves of a constructible sheaf of abelian groups (resp. A-modules) are constructible. (ii) Let φ : F → G be a morphism of constructible sheaves of abelian groups (resp. A-modules). Then ker φ, coker φ and im φ are constructible. (iii) The category of constructible sheaves of abelian groups (resp. Amodules) is an abelian category. Proof. Use 5.8.5 and the fact that subsheaves and quotient sheaves of a noetherian sheaf are noetherian. Proposition 5.8.7. Let X be a noetherian scheme, A a noetherian ring, and 0 → F 0 → F → F 00 → 0 an exact sequence of sheaves of abelian groups or A-modules. If F 0 and F 00 are constructible (resp. locally constant), then so is F .

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Proof. By 5.8.5, it suffices to treat the case of sheaves of A-modules. Suppose that F 0 and F 00 are constructible sheaves of A-modules. We Sn can find a decomposition X = i=1 Xi of X into finitely many locally closed subsets Xi and surjective etale morphisms Xi0 → Xi such that F 0 |Xi0 and F 00 |Xi0 are constant sheaves associated to A-modules Mi0 and Mi00 , respectively. It suffices to show that each F |Xi0 is locally constant. Let s001 , . . . , s00n ∈ Mi00 be a finite family of generators for Mi00 . We can find an etale covering {Uiα → Xi0 }α∈Iα of Xi0 such that each Uiα is connected and each s00j can be lifted to a section in F (Uiα ). We claim that F |Uiα is a constant sheaf. For any connected etale Uiα -scheme V , consider the commutative diagram 0 → F 0 (Uiα ) → F (Uiα ) → F 00 (Uiα ) → 0 ↓ ↓ ↓ 0 → F 0 (V ) → F (V ) → F 00 (V ) → 0. The horizontal lines are exact. Since F 0 |Xi0 and F 00 |Xi0 are constant, the first and the last vertical arrows are isomorphisms. It follows that F (Uiα ) → F (V ) is an isomorphism. So F |Uiα is constant. Therefore F |Xi0 is locally constant and F is constructible. Proposition 5.8.8. Let X be a noetherian scheme, A a noetherian ring, and F a torsion sheaf of abelian groups (resp. a sheaf of A-modules). Then constructible subsheaves of abelian groups (resp. A-modules) of F form a direct system and F is the direct limit of this system. Proof. If F1 and F2 are two constructible subsheaves of F , then so is F1 + F2 by 5.8.6 (ii). So constructible subsheaves of F form a direct S system. If F is a torsion sheaf, then F = n∈N ker (n : F → F ). Since ker (n : F → F ) are sheaves of Z/n-modules, to prove the proposition, it suffices to treat sheaves of A-modules. For any etale X-scheme U of finite type and any section s ∈ F (U ), we have a canonical morphism AU → F which maps the section 1 ∈ AU (U ) to s. The image of this morphism is a constructible subsheaf of F by 5.8.6 (i), and s is a section of it. So F is the direct limit of its constructible subsheaves. Proposition 5.8.9. Let X be a noetherian scheme, A a noetherian ring, F a sheaf of sets (resp. abelian groups, resp. A-modules) with finite (resp. finite, resp. finitely generated) stalks. Then F is locally constant if and only if for any x, y ∈ X with x ∈ {y} and any specialization morphism ey¯ → X ex¯ , the specialization map Fx¯ → Fy¯ is bijective. X

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Proof. The “only if” part is clear. Let us prove the “if” part. We say that two points x, y ∈ X can be connected by a path if there exists a sequence of points x1 = x, x2 , . . . , xn = y such that for any i ∈ {1, . . . , n − 1}, we have either xi ∈ {xi+1 }, or xi+1 ∈ {xi }. A subset Y of X is called path connected if any two points in Y can be connected by a path formed by a sequence of points in Y . Any irreducible component of X is path connected since any point in it can be connected to the generic point. If X1 and X2 are path connected subsets of X with nonempty intersection, then X1 ∪ X2 is path connected. It follows that connected components of X are path connected. Let x ∈ X, let s1¯x , . . . , sn¯x ∈ Fx¯ be all the elements of Fx¯ (resp. all elements of Fx¯ , resp. a finite family of generators of Fx¯ ), and let U be a connected quasi-compact etale neighborhood of x ¯ so that there exist sections s1 , . . . , sn ∈ F (U ) whose germs at x ¯ are s1¯x , . . . , sn¯x , respectively. Let G be the set {s1 , . . . , sn } (resp. the subgroup of F (U ) generated by {s1 , . . . , sn }, resp. the submodule of F (U ) generated by {s1 , . . . , sn }). We claim that when U is sufficiently small, the canonical map G → Fx¯ is bijective. In the case of sheaf of sets, this is clear. In the case of sheaf of abelian groups, {s1¯x , . . . , sn¯x } = Fx¯ is a group. So for any pair i, j ∈ {1, . . . , n}, there exists k(i, j) ∈ {1, . . . , n} such that si¯x − sj x¯ = sk(i,j),¯x . Replacing U by a sufficiently small connected quasi-compact etale neighborhood of x ¯ in U , we may assume si − sj = sk(i,j) . Then {s1 , . . . , sn } is a group. We have G = {s1 , . . . , sn } and G → Fx¯ is bijective. In the case of sheaf of A-modules, let (ai1 , . . . , ain ) (i = 1, . . . , m) be a finite family of generators for the kernel of the homomorphism An → Fx¯ ,

(a1 , . . . , an ) 7→ a1 s1¯x + · · · + an sn¯x .

We have ai1 s1¯x + · · · + ain sn¯x = 0

(i = 1, . . . , m).

Replacing U by a sufficiently small connected quasi-compact etale neighborhood of x ¯ in U , we may assume ai1 s1 + · · · + ain sn = 0 (i = 1, . . . , m). Then the kernel of the homomorphism An → F (U ),

(a1 , . . . , an ) 7→ a1 s1 + · · · + an sn

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is also generated by (ai1 , . . . , ain ) (i = 1, . . . , m). This implies that G ∼ = Fx¯ . We have a canonical morphism of sheaves G → F |U . By our choice of G, the map Gx¯ → (F |U )x¯ is bijective. Specialization maps for G are bijective. By assumption, specialization maps for F are also bijective. Moreover U is connected and hence path connected. These facts imply that Gy¯ → (F |U )y¯ is bijective for any y ∈ U . So G ∼ = F |U . Hence F is locally constant. Corollary 5.8.10. Let f : X → Y be a finite surjective radiciel morphism between noetherian schemes. Then the functor Y 0 7→ X ×Y Y 0 defines an equivalence from the category of etale covering spaces of Y to the category of etale covering spaces of X. In particular, if X is connected, and γ : s → X is a geometric point of X, then f induces an isomorphism ∼ =

f∗ : π1 (X, γ) → π1 (Y, f ◦ γ). Proof. By 5.8.1 (i) and 5.2.7 (iii), it suffices to show that the functor F → f ∗ F defines an equivalence from the category of locally constant sheaves of sets with finite stalks on Y to that on X. By 5.3.10, f ∗ defines an equivalence from the category of sheaves of sets on Y to that on X. Using 5.8.9, one shows that F is locally constant with finite stalks if and only if f ∗ F has the same property. Our assertions follows. Proposition 5.8.11. (i) Let f : X → Y be a finite morphism between noetherian schemes, A a noetherian ring, and F a sheaf of A-modules on X. Then F is constructible if and only if f∗ F is constructible. (ii) Suppose that X is a scheme of finite type over a field or over Z. Let A be a noetherian ring and let F be a constructible sheaf of A-modules on X. There exist noetherian integral schemes Xi (i = 1, . . . , m), finite morphisms pi : Xi → X, and finitely generated A-modules Mi such that we have a monomorphism

F ,→

m M

pi∗ Mi .

i=1

When A = Z/n, we can take Mi = Z/n.

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Proof. (i) Using 5.3.7, one can check that the canonical morphism f ∗ f∗ F → F is surjective. If f∗ F is constructible, then by 5.8.2 and 5.8.6 (i), F is constructible. Suppose F is constructible. By 5.8.3, to prove that f∗ F is constructible, it suffices to show that for any irreducible closed subset F of Y , there exists a nonempty open subset V of F such that (f∗ F )|V is locally constant. By 5.3.9, we have (f∗ F )|V ∼ = (fF )∗ (F |f −1 (F ) ) |V ,

where we put a reduced closed subscheme structure on F and fF : f −1 (F ) → F is the base change of f . Replacing f by fF , we may assume that Y is an integral scheme and prove that there exists a nonempty open subset V of Y such that (f∗ F )|V is locally constant. First consider the case where f is finite surjective and radiciel. Then f is a homeomorphism. There exists a nonempty open subset V of Y such that F |f −1 (V ) is locally constant. Using 5.8.9, one can check that (f∗ F )|V is locally constant. This proves our assertion under the assumption that f is finite surjective and radiciel. Now suppose that f is a finite morphism. Let K be the function field of Y . Then the base change X ×Y Spec K → Spec K of f is finite. So X ×Y Spec K is artinian, and hence n a ∼ (X ×Y Spec K)red = SpecLi , i=1

where Li are the residue fields of X ×Y Spec K at its generic points. For each i, let Ki be the separable closure of K in Li . We have a commutative diagram `n ∼ i=1 Spec Li = (X ×Y Spec K)red → X ×Y Spec K ↓ ↓ `n Spec K → Spec K. i i=1 By 1.10.10 (iv), (vi), (vii) and 2.3.7, there exists a nonempty open subset W in Y such that we have a commutative diagram p

U 0 → f −1 (W ) q ↓ ↓ fW r W0 → W

with p and q being finite surjective and radiciel, and r being finite etale, where fW is the base change of f . We have (f∗ F )|W ∼ = fW ∗ (F |f −1 (W ) ) ∼ = fW ∗ p∗ (F |U 0 ) = r∗ q∗ (F |U 0 ) ∼ = r! q∗ (F |U 0 ).

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We have shown that q∗ (F |U 0 ) is constructible. Using 5.8.5 (i) (c) and the fact that r! is an exact functor, one can show that r! q∗ (F |U 0 ) is constructible. So (f∗ F )|W is constructible. Hence there exists a nonempty open subset V of W such that (f∗ F )|V is locally constant. Sn (ii) Since F is constructible, there exists a decomposition X = i=1 Xi of X into finitely many locally closed subsets such that F |Xi are locally constant. Let ki : Xi → X be the immersions. For each i, there exists a surjective separated etale morphism of finite type fi : Xi0 → Xi such that 0 fi∗ ki∗ F is the constant sheaf associated to some A-module Mi . Let Xij be 0 the irreducible components of Xi with the reduced subscheme structures, let 0 e 0 be the normalizations lij : Xij → Xi0 be the closed immersions, and let X ij 0 of Xij . Since X is a scheme of finite type over a field or over Z, the e 0 → X 0 are finite ([Matsumura (1970)] (31 H) canonical morphisms πij : X ij ij Theorem 72) and surjective. Note that ki fi lij πij are quasi-finite separated morphisms and we have monomorphisms M M M F ,→ ki∗ fi∗ fi∗ ki∗ F ∼ ki∗ fi∗ Mi ,→ (ki fi lij πij )∗ Mi . = i

i

i,j

By the Zariski Main Theorem 1.10.13, there exist finite morphisms pij : e 0 ,→ Xij such that pij gij = Xij → X and open immersions gij : X ij ki fi lij πij . Replacing Xij by the closures of im (gij ) with the reduced closed subscheme structures, we may assume that Xij are integral schemes. Let eij be the normalizations of Xij . Then X eij are finite over Xij , and X e0 X ij e e can be regarded as open subschemes of Xij . Replacing Xij by Xij , we may assume that Xij are normal integral schemes. We claim that we then have gij∗ (Mi ) = Mi . Indeed, any connected object U of (Xij )et is normal and e 0 ×Xij U is a nonempty open subscheme of U and hence hence integral. X ij is irreducible and connected. It follows that e 0 ×Xij U ) = Mi = Mi (U ). (gij∗ (Mi ))(U ) = Mi (X ij

This proves our claim. We thus have M M M F ,→ (ki fi lij πij )∗ Mi ∼ pij∗ gij∗ Mi = pij∗ Mi . = i,j

i,j

i,j

L

So we have a monomorphism F ,→ i,j pij∗ Mi . When A = Z/n, any finitely generated A-module can be embedded into a free Z/n-module of finite rank. So we can take Mi = Z/n in our conclusion. One can also study constructible sheaves on non-noetherian schemes. Confer [SGA 4] IX 2.

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([SGA 4] VII 5, IX 2.) We use the same notation as in 1.10. We assume S0 is quasi-compact and quasi-separated. Then Sλ and S are also quasi-compact and quasif separated. By 5.2.8, we can work with sheaves on (Sλ )fet and Set instead of sheaves on (Sλ )et and Set . Define the category S of sheaves on (Sλ , uλ ) as follows: An object in S is a family (Fλ , ψλµ )λ∈I , where Fλ is a sheaf of abelian groups on (Sλ )fet for each λ ∈ I, ψλµ : u∗λµ (Fλ ) → Fµ

is a morphism of sheaves for each pair λ ≤ µ, and ψλν = ψµν ◦ u∗µν (ψλµ )

for each triple λ ≤ µ ≤ ν. We call such an object a sheaf on (Sλ , uλµ ). We often denote it by (Fλ ) for simplicity. A morphism from a sheaf (Fλ , ψλµ ) 0 to another sheaf (Fλ0 , ψλµ ) on (Sλ , uλµ ) is a family (θλ ), where θλ : Fλ → 0 Fλ are morphisms of sheaves such that for any pair λ ≤ µ, we have 0 θµ ◦ ψλµ = ψλµ ◦ u∗λµ (θλ ).

One can show that S is an abelian category. We can also define the category P of presheaves on (Sλ , uλµ ). A presheaf on (Sλ , uλµ ) is a family (Fλ , ψλµ )λ∈I , where Fλ is a presheaf of abelian groups on (Sλ )fet for each λ ∈ I, ψλµ : uP λµ (Fλ ) → Fµ

is a morphism of presheaves for each pair λ ≤ µ, and ψλν = ψµν ◦ uP µν (ψλµ )

for each triple λ ≤ µ ≤ ν. For any λ and any sheaf Fλ on Xλ , let F λ = ((F λ )µ , ψµν )µ∈I be the sheaf on (Sλ , uλµ ) defined as follows: If µ ≥ λ, we define the µ-component (F λ )µ of F λ to be u∗λµ Fλ . Otherwise, we define the (F λ )µ to be 0. For any µ ≤ ν, we define ψµν : u∗µν (F λ )µ → (F λ )ν to be 0 if (F λ )µ = 0, and we define it to be the canonical isomorphism u∗µν (u∗λµ Fλ ) ∼ = u∗λν Fλ if µ ≥ λ. For any sheaf G = (Gλ ) on (Sλ , uλµ ), we have a one-to-one correspondence Hom(F λ , G ) = Hom(Fλ , Gλ ).

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So the functor SSλ → S ,

Fλ 7→ F λ

S → SSλ ,

(Gλ ) 7→ Gλ .

is left adjoint to the functor

For each λ ∈ I, let Jλ be a set of objects πλ : Uλ → Sλ in (Sλ )fet with the property that Uλ are affine and their images in Sλ are also affine, and any object in (Sλ )fet with this property is isomorphic to an object in Jλ . When λ goes over I and πλ goes over Jλ , πλ! Z form a set of generators for the category S . By the same argument as the proof of [Fu (2006)] 2.1.6 or [Grothendieck (1957)] 1.10.1, S has enough injective objects. Since the functor Fλ 7→ F λ is exact, if (Iλ ) is an injective sheaf on (Sλ , uλµ ), then Iλ is an injective sheaf on Sλ for each λ. Given a presheaf (Fλ ) on (Sλ , uλµ ), define a presheaf F on S as follows. f For any object U in Set , by 1.10.9 (iii) and 2.3.7, we can find λU ∈ I and an f object UλU in (SλU )et such that we have an S-isomorphism U ∼ = UλU ×SλU S. Define F (U ) = lim Fλ (Uλ ), −→ λ≥λU

where Uλ = UλU ×SλU Sλ for any λ ≥ λU . Given a morphism U → V in f Set , we can find λ0 ≥ λU , λV and an Sλ0 -morphism Uλ0 → Vλ0 inducing the given S-morphism U → V . We define the restriction F (V ) → F (U ) to be the direct limit of the restrictions Fλ (Vλ ) → Fλ (Uλ ) for λ ≥ λ0 . We thus get a presheaf F . Suppose that (Fλ ) is a sheaf on (Sλ , uλµ ), then F f f . Indeed, for any etale covering U = {Ui → U }i∈I in Set , we is a sheaf on Set can find λU ≥ λU , λUi and an etale covering {UiλU → UλU }i∈I in (SλU )fet inducing the etale covering U. For any λ ≥ λU , we have the canonical exact sequence Y Y 0 → Fλ (Uλ ) → Fλ (Uiλ ) → Fλ (Uiλ ×Uλ Ujλ ). i

i,j

Taking direct limit, we get an exact sequence Y Y 0 → F (U ) → F (Ui ) → F (Ui ×U Uj ). i

i,j

So F is a sheaf.

Lemma 5.9.1. Let (Pλ ) be a presheaf on (Sλ , uλµ ). Then (Pλ# ) is a sheaf on (Sλ , uλµ ). Let P (resp. F ) be the presheaf (resp. sheaf ) on S corresponding to (Pλ ) (resp. (Pλ# )) defined as above. We have P # ∼ = F.

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Proof.

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f For any sheaf G on Set , we have # ∼ Hom(P , G ) = Hom(P, G ) ∼ Hom(Pλ , uλ∗ G ) = lim ←− λ

∼ Hom(Pλ# , uλ∗ G ) = lim ←− λ

∼ = Hom(F , G ).

Proposition 5.9.2. Given a sheaf (Fλ ) on (Sλ , uλµ ), define the sheaf F on S as above. (i) We have F ∼ u∗ F . = lim −→λ λ λ (ii) If each Fλ is flasque, then F is flasque. (iii) We have H q (S, F ) ∼ H q (Sλ , Fλ ) for all q. = lim −→λ Proof. f , we have (i) For any sheaf G on Set Hom(F , G ) ∼ Hom(Fλ , uλ∗ G ) = lim ←− λ

∼ Hom(u∗λ Fλ , G ) = lim ←− λ

∼ u∗ F , G ). = Hom(lim −→ λ λ λ

f (ii) Let U = {Ui → U }i∈I be an etale covering in Set . We can find λ ≥ λU , λUi such that there exists an etale covering Uλ = {Uiλ → Uλ }i∈I in (Sλ )fet inducing U. It is clear that ˇ q (Uλ , Fλ ). ˇ q (U, F ) ∼ H H = lim −→ λ≥λU

ˇ q (Uλ , Fλ ) = 0 for all q ≥ 1. It follows that Since Fλ are flasque, we have H ˇ q (U, F ) = 0 for all q ≥ 1. So F is flasque. H (iii) Let 0 → (Iλ0 ) → (Iλ1 ) → · · · be an injective resolution of (Fλ ) in the category of sheaves on (Sλ , uλµ ). Then for each λ, Iλ· is an injective resolution of Fλ . By (i) and (ii), limλ u∗λ Iλ· is a flasque resolution of F . So we have −→ H q (S, F ) ∼ u∗ I · )) = H q (Γ(S, lim −→ λ λ λ

∼ Γ(Sλ , Iλ· )) = H q (lim −→ λ

∼ H q (Γ(Sλ , Iλ· )) = lim −→ λ

∼ H q (Sλ , Fλ ). = lim −→ λ

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Corollary 5.9.3. Let F0 be a sheaf on S0 , and let Fλ and F be the inverse images of F0 on Sλ and S, respectively. We have H q (S, F ) ∼ = limλ H q (Sλ , Fλ ) for all q. −→ Corollary 5.9.4. Let F be a sheaf on S and let Fλ = uλ∗ F . We have H q (S, F ) ∼ H q (Sλ , Fλ ) for all q. = lim −→λ Corollary 5.9.5. Let f : X → Y be a quasi-compact and quasi-separated morphism, F a sheaf on X, y a point on Y , and Yey¯ the strict localization of Y at y¯. Then for all q, we have (Rq f∗ F )y¯ ∼ = H q (X ×Y Yey¯, F | e ). X×Y Yy¯

Proof. have

Let {Vλ } be the family of affine etale neighborhood of y¯ in Y . We Yey¯ = Spec (lim Γ(Vλ , OVλ )). −→ λ

For each q, Rq f∗ F is the sheaf associated to the presheaf V 7→ H q (X ×Y V, F |X×Y V )

for any V ∈ ob Yet . So we have ∼ lim H q (X ×Y Vλ , F |X× V ). (Rq f∗ F )y¯ = Y λ −→ λ

By 5.9.3, we have

lim H q (X ×Y Vλ , F |X×Y Vλ ) ∼ = H q (X ×Y Yey¯, F |X×Y Yey¯ ). −→ λ

Our assertion follows.

Corollary 5.9.6. Let X0 be an S0 -scheme, (Xλ , vλµ ) another inverse system of X0 -schemes such that Xλ are affine over X0 for all λ ∈ I, X = limλ Xλ , v : X → Xλ and vλµ : Xµ → Xλ (λ ≤ µ) the projec←− tions, fλ : Xλ → Sλ quasi-compact quasi-separated S0 -morphisms satisfying fλ vλµ = uλµ fµ for any λ ≤ µ, and f : X → S the morphism induced by (fλ ) by passing to limit. (i) Let (Fλ ) be a sheaf on (Xλ , vλµ ), and let F = limλ vλ∗ Fλ . Then we −→ have R q f∗ F ∼ u∗ Rq fλ∗ Fλ = lim −→ λ λ

for all q. (ii) Let (Gλ ) be a sheaf on (Sλ , vλµ ), and let G = limλ u∗λ Gλ . Then we −→ have ∼ lim v ∗ f ∗ Gλ . f ∗G = −→ λ λ λ

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Proof. f (i) For each q, Rq f∗ F (resp. Rq fλ∗ Fλ ) is the sheaf on Set (resp. (Sλ )fet ) associated to the presheaf V 7→ H q (X ×S V, F )

(resp. Vλ 7→ H q (Xλ ×Sλ Vλ , Fλ ))

f for any V ∈ ob Set (resp. Vλ ∈ ob (Sλ )fet ). By 5.9.2 (iii), we have H q (X ×S V, F ) ∼ H q (Xλ ×Sλ Vλ , Fλ ). = lim −→ λ≥λV

Our assertion then follows from 5.9.1. (ii) We have f ∗G ∼ u∗ G ) = f ∗ (lim −→ λ λ λ

∼ f ∗ u∗λ Gλ = lim −→ λ

∼ v∗ f ∗ G . = lim −→ λ λ λ λ

Corollary 5.9.7. Let f : X → S be an integral morphism and let F be a sheaf on X. Then Rq f∗ F = 0 for any q ≥ 1. Moreover, for any Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f g S 0 → S,

we have a canonical isomorphism

∼ =

g ∗ f∗ F → f∗0 g 0∗ F . Proof. The canonical morphism g ∗ f∗ F → f∗0 g ∗ F is obtained by adjunction from the composite of the canonical morphisms f∗ (adj)

→ f∗ g∗0 g 0∗ F ∼ = g∗ f∗0 g 0∗ F . ∼ f 0 g ∗ F if f is a finite morphism. By 5.3.9, we have g ∗ f∗ F = ∗ To prove 5.9.7, the problem is local on S, and we may assume S = Spec A is affine. Since X is integral over S, it is also affine. Let {Aλ } be the family subalgebras of Γ(X, OX ) finitely generated over A, let Sλ = Spec Aλ , and let uλ : X → Sλ and fλ : Sλ → S be the canonical morphisms. Using 5.9.2 (i), one can show F ∼ u∗ u F . Applying 5.9.6 to the morphism = lim −→λ λ λ∗ (fλ ) from the inverse system (Sλ ) to the constant inverse system (S), we get ∼ lim Rq fλ∗ (uλ∗ F ). R q f∗ F = −→ f∗ F

λ

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Each fλ is a finite morphism. By 5.7.4, we have Rq fλ∗ (uλ∗ F ) = 0 for any q ≥ 1. So we have

R q f∗ F = 0

for any q ≥ 1. In the case q = 0, we get f∗ F ∼ f u F. = lim −→ λ∗ λ∗ λ

Fix notation by the following commutative diagram: u0

f0

λ λ X ×S S 0 → Sλ × S S 0 → S0 0 g ↓ gλ ↓ g ↓

X

u

λ →

Sλ

fλ

→ S.

Since F ∼ u∗ u F , we have = lim −→λ λ λ∗ g 0∗ F ∼ u0∗ g ∗ u F . = lim −→ λ λ λ∗ λ (fλ0 )

Applying 5.9.6 to the morphism from the inverse system (Sλ ×S S 0 ) to 0 the constant inverse system (S ), we get f∗0 g 0∗ F ∼ f 0 g∗ u F . = lim −→ λ∗ λ λ∗ λ

So we have g ∗ f∗ F ∼ f u F) = g ∗ (lim −→ λ∗ λ∗ λ

∼ g ∗ fλ∗ uλ∗ F = lim −→ λ

∼ f 0 g∗ u F = lim −→ λ∗ λ λ∗ λ

∼ = f∗0 g 0∗ F ,

where for the third isomorphism, we apply 5.3.9 to the finite morphisms fλ . Lemma 5.9.8. Suppose that S0 , Sλ , and S are noetherian schemes, and A is a noetherian ring. (i) Let F0 be a sheaf of A-modules on S0 , Fλ and F its inverse images on Sλ and S, respectively. If F0 is a constructible sheaf of A-modules, then for any sheaf of A-modules (Gλ ) on (Sλ , uλµ ), we have a one-to-one correspondence lim HomA (Fλ , Gλ ) → HomA (F , G ), −→ λ

where G = limλ u∗λ Gλ . −→

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(ii) Let F0 and G0 be constructible sheaves of A-modules on S0 , Fλ and Gλ (resp. F and G ) their inverse images on Sλ (resp. S), φ0 : F0 → G0 a morphism, and φλ : Fλ → Gλ and φ : F → G the morphisms induced by φ0 . If φ is an isomorphism, then φλ are isomorphisms for sufficiently large λ. (iii) For any constructible sheaf of A-modules F on S, we can find a sufficiently large λ and a constructible sheaf of A-modules Fλ on Sλ such that F ∼ = u∗λ F . (iv) Let φ0

ψ0

F0 → G0 → H0 be a sequence of constructible sheaves on S0 . If its inverse image φ

ψ

F →G →H on S is exact, then for sufficiently large λ, its inverse image φλ

ψλ

Fλ → Gλ → Hλ on Sλ is exact. Proof. (i) Any morphism φλ : Fλ → Gλ induces a morphism F → G by taking the composite F = u∗λ Fλ

u∗ λ (φλ )

→

u∗λ Gλ → lim u∗λ Gλ =G . −→ λ

We thus get maps HomA (Fλ , Gλ ) → HomA (F , G ). They are compatible with each other and induce a map lim HomA (Fλ , Gλ ) → HomA (F , G ). −→ λ

Let us prove that the last map is bijective. For any etale S0 -scheme U0 of finite type, let Uλ = U0 ×S0 Sλ and U = U0 ×S0 S. We have lim HomA (AUλ , Gλ ) ∼ G (U ) = lim −→ −→ λ λ λ

λ

∼ = G (U ) ∼ = HomA (AU , G ).

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In general, by 5.8.5, we have an exact sequence AU0 → AV0 → F0 → 0 for some etale S0 -schemes U0 and V0 of finite type. Consider the commutative diagram 0 → limλ HomA (Fλ , Gλ ) → limλ HomA (AVλ , Gλ ) → HomA (AUλ , Gλ ) −→ −→ ↓ ↓ ↓ 0→ HomA (F , G ) → HomA (AV , G ) → HomA (AU , G ). The horizontal lines are exact. The last two vertical arrows are bijective. It follows that the first vertical arrow is bijective. (ii) follows from (i). (iii) We can find an exact sequence AU → AV → F → 0 for some etale S-schemes U and V of finite type. We can find λ and etale Sλ -schemes Uλ and Vλ of finite type such that U ∼ = Uλ ×Sλ S and V ∼ = Vλ ×Sλ S. By (i), taking λ sufficiently large, we can find a morphism AUλ → AVλ inducing the morphism AU → AV . Let Fλ be the cokernel of AUλ → AVλ . Then F ∼ = u∗λ Fλ . (iv) We have ψφ = 0. By (i), we have ψλ φλ = 0 for sufficiently large λ. Let Kλ (resp. K ) be the kernel of ψλ (resp. ψ), and let Lλ (resp. L ) be the image of φλ (resp. φ). We have L ∼ = K . By (ii), we have Lλ ∼ = Kλ for sufficiently large λ. Lemma 5.9.9. Suppose that S0 , Sλ and S are noetherian schemes, and A is a noetherian ring. Let (Iλ ) be an injective sheaf of A-modules on (Sλ , uλµ ), and let I = limλ u∗λ Iλ . Then I is an injective of A-modules −→ on S. Proof. Sheaves of the form AU with U being etale S-schemes of finite type form a family of generators for the category of sheaves of A-modules on S. To prove that I is injective, it suffices to show that for any subsheaf F of AU , any morphism F → I can be extended to a morphism AU → I . By 5.8.6, F is constructible. By 5.9.8, for sufficiently large λ, we can find an etale Sλ -scheme Uλ of finite type such that U ∼ = Uλ ×Sλ S, a subsheaf Fλ of AUλ such that F ∼ = u∗λ Fλ , and a morphism Fλ → Iλ inducing the given morphism F → I . Since Iλ is necessarily injective, the morphism Fλ → Iλ can be extended to a morphism AUλ → Iλ . It follows that F → I can be extended to a morphism AU → I .

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Proposition 5.9.10. Suppose that S0 , Sλ and S are noetherian schemes, and A is a noetherian ring. Let F0 be a sheaf of A-modules on S0 , Fλ and F its inverse images on Sλ and S, respectively. If F is constructible, then for any sheaf of A-modules (Gλ ) on (Sλ , uλµ ), we have ExtqA (Fλ , Gλ ), ExtqA (F , G ) ∼ = lim −→ λ

E xtqA (F , G )

∼ u∗ E xtqA (Fλ , Gλ ) = lim −→ λ λ

for all q, where G = limλ u∗λ Gλ . −→ Proof.

Let 0 → (Iλ0 ) → (Iλ1 ) → · · ·

be a resolution of (Gλ ) by injective sheaves of A-modules on (Sλ , uλµ ). By 5.9.9, limλ u∗λ Iλ· is an injective resolution of G , and each Iλ· is an injective −→ resolution of Gλ . By 5.9.8, we have HomA (Fλ , Iλ· ). HomA (F , lim u∗λ Iλ· ) ∼ = lim −→ −→ λ

λ

So we have ExtqA (F , G ) ∼ u∗ I · )) = H q (HomA (F , lim −→ λ λ λ

∼ HomA (Fλ , Iλ· )) = H q (lim −→ λ

∼ H q (HomA (Fλ , Iλ· )) = lim −→ λ

∼ ExtqA (Fλ , Gλ ). = lim −→ λ

q

Note that E xt (F , G ) is the sheaf associated to the presheaf U 7→ f ExtqA (F |U , G |U ) for any U ∈ ob Set . We have ExtqA (F |U , G |U ) ∼ ExtqA (Fλ |Uλ , Gλ |Uλ ). = lim −→ λ≥λU

Since E xtq (Fλ , Gλ ) is the sheaf associated to the presheaf Uλ 7→ ExtqA (Fλ |Uλ , Gλ |Uλ ) for any Uλ ∈ ob (Sλ )fet , we have E xtqA (F , G ) ∼ u∗ E xtqA (Fλ , Gλ ) = lim −→ λ λ

by 5.9.1 and 5.9.2.

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Corollary 5.9.11. Let S be a noetherian scheme, A a noetherian ring, s ∈ S, Ses¯ the strict henselization of S at s¯, and F and G sheaves of Amodules on S. If F is constructible, then q (E xtqA (F , G ))s¯ ∼ = ExtA (F |Ses¯ , G |Ses¯ ).

Proof. have

Let (Vλ ) be the family of affine etale neighborhoods of s¯ in S. We Ses¯ = Spec (lim Γ(Vλ , OVλ )). −→ λ

E xtqA (F , G )

Since is the sheaf associated to the presheaf V ExtqA (F |V , G |V ), we have

7→

(E xtqA (F , G ))s¯ ∼ ExtqA (F |Vλ , G |Vλ ). = lim −→ λ

By 5.9.10, we have q lim ExtqA (F |Vλ , G |Vλ ) ∼ = ExtA (F |Ses¯ , G |Ses¯ ). −→ λ

Our assertion follows.

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Chapter 6

Derived Categories and Derived Functors

In this chapter, covariant functors between categories are simply called functors.

6.1

Triangulated Categories

([Beilinson, Bernstein and Deligne (1982)] 1.1, [Hartshorne (1966)] I 0–2, [SGA 4] XVII 1.1, [SGA 4 21 ] C.D. I 1.) A triangulated category C is an additive category provided with an automorphism T : C → C , called the translation functor, and a collection of sextuples (X, Y, Z, u, v, w), called distinguished triangles and also denoted by u

v

w

X → Y → Z →,

where X, Y and Z are objects in C , u : X → Y , v : Y → Z and w : Z → T (X) are morphisms in C , such that the following axioms hold: (TR1) Any morphism u : X → Y in C can be extended to a distinguished triangle u

X →Y →Z→.

The sextuple (X, X, 0, idX , 0, 0) is a distinguished triangle for any object X in C . Any sextuple isomorphic to a distinguished triangle is a distinguished triangle. Here a morphism from a sextuple (X, Y, Z, u, v, w) to another f

g

h

sextuple (X 0 , Y 0 , Z 0 , u0 , v 0 , w0 ) is a triple (X → X 0 , Y → Y 0 , Z → Z 0 ) of morphisms in C such that the following diagram commutes: u

v

w

u0

v0

w0

X → Y → Z → T (X) f ↓ g ↓ h↓ ↓ T (f )

X 0 → Y 0 → Z 0 → T (X 0 ). 267

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We define composites of morphisms of sextuples in the obvious way. An isomorphism of sextuples is a morphism with a two-sided inverse. (TR2) A sextuple (X, Y, Z, u, v, w) is a distinguished triangle if and only if (Y, Z, T (X), v, w, −T (u)) is a distinguished triangle. (TR3) Given two distinguished triangles X → Y → Z → and X 0 → Y 0 → Z 0 →, and two morphisms f : X → X 0 and g : Y → Y 0 such that the diagram X→Y f ↓ ↓g X0 → Y 0 commutes, there exists a morphism h : Z → Z 0 such that (f, g, h) is a morphism of distinguished triangles. (TR4) Given two morphisms u : X → Y and v : Y → Z, and three triangles u

j

X → Y → Z 0 →,

v

i

Y → Z → X 0 →,

vu

X → Z → Y 0 →,

there exist morphisms f : Z 0 → Y 0 and g : Y 0 → X 0 such that (idX , v, f ) and (u, idZ , g) are morphisms of distinguished triangles, and the sextuple (Z 0 , Y 0 , X 0 , f, g, T (j) ◦ i) is a distinguished triangle. Let C 0 and C be triangulated categories with the translation functors T and T , respectively. An additive covariant (resp. contravariant) functor F : C 0 → C is called exact if there exists an isomorphism of functors ∼ ∼ = = φ : F ◦ T 0 → T ◦ F (resp. φ : F ◦ T 0−1 → T ◦ F ) such that for any distinguished triangle (X 0 , Y 0 , Z 0 , u0 , v 0 , w0 ) in C 0 , the sextuple 0

(resp.

(F (X 0 ), F (Y 0 ), F (Z 0 ), F (u0 ), F (v 0 ), φX 0 ◦ F (w0 ))

(F (Z 0 ), F (Y 0 ), F (X 0 ), F (v 0 ), F (u0 ), φZ 0 ◦ F (T −1 (w0 ))))

is a distinguished triangle in C . Let I be a finite set, : I → {±1} a map, C a triangulated category with the translation functor T , and Ci (i ∈ I) triangulated categories with the Q translation functors Ti , and F : i Ci → C an additive functor covariant (resp. contravariant) in the i-th component if (i) = 1 (resp. (i) = −1). Q Q (i) For any i ∈ I, denote by (Ti ) the functor i Ci → i Ci induced by (i) Ti and idCj for j 6= i. We say that F is an exact functor if the following conditions hold:

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(i)

(a) There exist isomorphisms of functors φi : F ◦ (Ti such that for any i 6= j, the diagram (i)

F ◦ (Ti

(j)

) ◦ (Tj

(j)

) = F ◦ (Tj φi ↓ (j)

T ◦ F ◦ (Tj

(i)

) ◦ (Ti

∼ =

) → T ◦ F (i ∈ I)

φj

(i)

) → T ◦ F ◦ (Ti ↓ φi

)

φj

)

→ T ◦T ◦F

is anti-commutative, that is, φi φj = −φj φi . Q (b) With respect to each component of i Ci , F is an exact functor.

Proposition 6.1.1. Let C be a triangulated category. u v w (i) For any distinguished triangle X → Y → Z → in C , we have vu = 0 and wv = 0. u v w (ii) For any distinguished triangle X → Y → Z → and any object W in C , the following sequences are exact · · ·→ Hom(W,T i X) → Hom(W,T i Y) → Hom(W,T i Z) → Hom(W,T i+1 X) →· · · ,

· · ·← Hom(T i X,W) ← Hom(T i Y,W) ← Hom(T i Z,W) ← Hom(T i+1 X,W) ←· · · . (iii) In the axiom (TR3), if f and g are isomorphisms, then so is h.

Proof. (i) The pair of morphisms (idX , u) gives rise to a commutative diagram id

X →X X idX ↓ ↓u u X → Y. By (TR3), it can be extended to a morphism from the distinguished triangle id

X →X X → 0 → to the distinguished triangle u

v

w

X →Y →Z→. So we have a commutative diagram id

X →X X → 0 → T (X) idX ↓ ↓u ↓ ↓idT (X) u v w X → Y → Z → T (X). It follows that vu = 0. By (TR2), v

w

−T (u)

Y → Z → T (X) →

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is also a distinguished triangle. It follows that wv = 0 from the discussion above. (ii) It suffices to prove that Hom(W, X) → Hom(W, Y ) → Hom(W, Z),

Hom(X, W ) ← Hom(Y, W ) ← Hom(Z, W )

are exact and then apply (TR2) repeatedly. By (i), the composites of morphisms in these two sequences are zero. Suppose that f lies in the kernel of Hom(W, Y ) → Hom(W, Z). We have a commutative diagram W → 0 ↓ ↓ v Y → Z.

f

By (TR2) and (TR3), there exists h ∈ Hom(W, X) making the following diagram commute: id

W →W → 0 → h↓ f ↓ ↓ u v w X → Y →Z→. We then have f = uh, and f lies in the image of Hom(W, X) → Hom(W, Y ). This proves that the first sequence is exact. Similarly, one can prove that the second sequence is exact. (iii) Consider the commutative diagram Hom(Z 0 ,X) → Hom(Z 0 ,Y ) → Hom(Z 0 ,Z) → Hom(Z 0 ,T (X)) → Hom(Z 0 ,T (Y )) ↓

↓

↓

↓

↓

Hom(Z 0 ,X 0 ) → Hom(Z 0 ,Y 0 ) → Hom(Z 0 ,Z 0 ) → Hom(Z 0 ,T (X 0 )) → Hom(Z 0 ,T (Y 0 )),

where the vertical arrows are induced by f, g, h, T (f ), T (g), respectively. By (ii), the horizontal lines are exact. If f and g are isomorphisms, then by the five lemma, the morphism Hom(Z 0 , Z) → Hom(Z 0 , Z 0 ) induced by h is an isomorphism. So there exists h0 : Z 0 → Z such that hh0 = idZ 0 . Similarly one can prove that h has a left inverse. Hence h is an isomorphism. Let A be an additive category and let K(A ) be the category whose objects are complexes of objects in A , and whose morphisms are homotopy classes of morphisms of complexes. A complex X · is bounded below (resp. bounded above, resp. bounded) if X n = 0 for n 0 (resp. n 0, resp.

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n 0 and n 0). Denote by K + (A ), K − (A ) and K b (A ) the full subcategories of K(A ) consisting of bounded below, bounded above, and bounded complexes, respectively. Define the translation functor T : K(A ) → K(A ) as follows. For any complex X · ∈ ob K(A ), T (X ·) is the complex defined by T (X · )i = X i+1 ,

diT (X · ) = −di+1 X· .

For any morphism u· : X · → Y · , define T (u· ) : T (X · ) → T (Y · ) by T (u·)i = ui+1 .

We also write X · [1] for T (X · ) and write X · [n] for T n (X) for any integer n. For any morphism of complexes u : X · → Y · in K(A ), define the mapping cone C(u)· of u to be the complex defined by C(u)p = X p+1 ⊕ Y p ,

d : C(u)p → C(u)p+1 ,

(xp+1 , yp ) 7→ (−dxp+1 , u(xp+1 ) + dyp ).

One can check that dd = 0. We have canonical morphisms Y · → C(u)· and C(u)· → X · [1]. Any sextuple isomorphic to u

X · → Y · → C(u)· →

is called a distinguished triangle in K(A ). One can show that K(A ) is a triangulated category, and the subcategories K + (A ), K − (A ) and K b (A ) are also triangulated categories. Moreover, we have the following. Proposition 6.1.2. Suppose that A is an abelian category. Let X· → Y · → Z· → be a distinguished triangle in K(A ). Then the sequence · · · → H i (X · ) → H i (Y · ) → H i (Z · ) → H i+1 (X · ) → · · · is exact. Suppose that A is an abelian category. An abelian subcategory A 0 of A is called thick if for any exact sequence 0 → X 0 → X → X 00 → 0

in A with X 0 , X 00 ∈ ob A 0 , we have X ∈ ob A 0 . For any thick abelian subcategory A 0 of A , we defined KA 0 (A ) to be the full subcategory of K(A ) consisting of the complexes X · in K(A ) such that H q (X · ) are objects in A 0 for all q. By 6.1.2, KA 0 (A ) is a triangulated subcategory + of K(A ). Similarly, we can define triangulated subcategories KA 0 (A ), − b KA 0 (A ) and KA 0 (A ).

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6.2

Etale Cohomology Theory

Derived Categories

([Hartshorne (1966)] I 3–4, [SGA 4 21 ] C.D. I 2, II 1.) Let C be a category. A family S of morphisms in C is called a multiplicative system if it satisfies the following conditions: (a) For any f, g ∈ S, we have f g ∈ S whenever the composite f g is defined. Identity morphisms are in S. (b) Any diagram X →Y ↓ Z

(resp.

s

Y ↓t Z → W)

with s ∈ S (resp. t ∈ S) can be completed to a commutative diagram X → Y s↓ ↓t Z →W

with t ∈ S (resp. s ∈ S). (c) Let f, g : X → Y be two morphisms. There exists s ∈ S such that sf = sg if and only if there exists t ∈ S such that f t = gt. Let C be a category and let S be a multiplicative system of morphisms in C . For any X ∈ ob C , define a category X − (resp. X + ) as follows: s s Objects of X − (resp. X + ) are morphisms X 0 → X (resp. X → X 0 ) in S. We also denote these objects by X 0 for simplicity. Given two objects s s Xi0 →i X (resp. X →i Xi0 ) (i = 1, 2) in X − (resp. X + ), morphisms from s1 to s2 are morphisms f : X10 → X20 in C such that s2 f = s1 (resp. f s1 = s2 ). One can verify (X − )◦ and X + satisfy the conditions (I2) and (I3) in 2.7, where (X − )◦ is the opposite category of X − . Define a category S −1 C as follows: S −1 C has the same family of objects as C . For any X, Y ∈ ob S −1 C , HomC (−, Y ) is a contravariant functor on X − , and we define HomS −1 C (X, Y ) =

lim −→

HomC (X 0 , Y ).

X 0 ∈ob (X − )◦

If an element in HomS −1 C (X, Y ) is represented by an element in f : X 0 → Y in HomC (X 0 , Y ) for some object s : X 0 → X in X − , we denote this element by the diagram X0 s↓ X

f

&

Y.

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Given two morphisms X0 s↓ X

f

&

Y0 t↓ g & Y Z

Y,

in S −1 C , find a commutative diagram X 00 t0 ↓ X0

f0

f

&

Y0 & ↓t Y

such that t0 : X 00 → X 0 lies in S. The composite of the above two morphisms is defined to be the morphism represented by the diagram X 00 st0 ↓ gf 0 & X Z. One can verify that the composite of morphisms in S −1 C is well-defined, that is, independent of the choice of diagrams representing morphisms. We have HomS −1 C (X, Y ) =

lim −→

HomC (X 0 , Y )

X 0 ∈ob (X − )◦

=

lim −→ 0

HomC (X, Y 0 )

Y ∈ob Y +

=

lim −→

HomC (X 0 , Y 0 ).

X 0 ∈ ob (X − )◦ , Y 0 ∈ ob Y +

We call S −1 C the localization of C with respect to S. We have a canonical functor Q : C → S −1 C . It transforms morphisms of C in S into isomorphisms in S −1 C . Proposition 6.2.1. Let C and D be categories, S a multiplicative system of morphisms in C , F : C → D a functor that transforms morphisms of C lying in S into isomorphisms in D, and Q : C → S −1 C the canonical functor. Then there exists a unique functor G : S −1 C → D such that F = G ◦ Q. Proposition 6.2.2. Let C be a triangulated category with the translation functor T and let S be a multiplicative system of morphisms in C . Suppose the following two conditions hold:

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(a) For any s ∈ S, we have T (s) ∈ S. (b) In the axiom (TR3), if f, g ∈ S, then h ∈ S. By (a), T induces an automorphism on S −1 C . A sextuple in S −1 C is called a distinguished triangle if it is isomorphic in S −1 C to a sextuple coming from a distinguished triangle in C . Then S −1 C is a triangulated category, and the canonical functor Q : C → S −1 C is exact. Let A be an abelian category, and let X · and Y · be two complexes of objects in A . A morphism u : X · → Y · in K(A ) is called a quasiisomorphism if u induces isomorphisms H i (X · ) → H i (Y · ) for all i. By 6.1.2, this is equivalent to saying that the mapping cone C(u)· is acyclic, that is, H i (C(u)· ) = 0 for all i. Proposition 6.2.3. Let A be an abelian category, let A 0 be a thick abelian subcategory of A , and let S be the family of quasi-isomorphisms in K(A ) + − b (resp. K + (A ), K − (A ), K b (A ), KA 0 (A ), KA 0 (A ), KA 0 (A ), KA 0 (A )). Then S is a multiplicative system satisfying the conditions (a) and (b) in 6.2.2. Proof. It is clear that if s : Y → Z and t : X → Y are quasiisomorphisms, then so is st. Given a diagram u X →Y s ↓ Z with s ∈ S, by (TR1) and (TR2), we can choose a distinguished triangle v

s

C→X→Z→.

Since s is a quasi-isomorphism, C is acyclic by 6.1.2. By (TR1), we can choose a distinguished triangle uv

t

C→Y →W →.

Then t is a quasi-isomorphism. By (TR3), the pair (idC , u) can be extended v s uv t to a morphism of triangles (idC , u, g) from C → X → Z → to C → Y → W →. Then we have a commutative diagram u X → Y s↓ ↓t g Z →W with t ∈ S. Similarly any diagram Y ↓t Z→W

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with t ∈ S can be completed to a commutative diagram X → Y ↓ ↓t Z →W

s

with s ∈ S. Let f : X → Y be a morphism and let s : Y → Y 0 be a quasiisomorphism such that sf = 0. By (TR1) and (TR2), we can choose a distinguished triangle u

s

C →Y →Y0 →. Since s is a quasi-isomorphism, C is acyclic. Since sf = 0, by 6.1.1 (ii), there exists a morphism v : X → C such that f = uv. By (TR1) and (TR2), we can choose a distinguished triangle t

v

X0 → X → C → . Since C is acyclic, t is a quasi-isomorphism. We have f t = uvt = 0. Similarly, one can prove that if there exists a quasi-isomorphism t such that f t = 0, then there exists a quasi-isomorphism s such that sf = 0. It is clear that S satisfies the condition (a) in 6.2.2. It follows from 6.1.2 and the five lemma that S satisfies the condition (b). Let A be an abelian category, A 0 a thick abelian subcategory of A , and S the family of quasi-isomorphisms in K(A ) (resp. K + (A ), K − (A ), K b (A ), + − b KA 0 (A ), KA 0 (A ), KA 0 (A ), KA 0 (A )).

The category S −1 K(A ) (resp. S −1 K + (A ), S −1 K − (A ), S −1 K b (A ), + −1 − b S −1 KA 0 (A ), S −1 KA KA 0 (A ), S −1 KA 0 (A )) 0 (A ), S

is called a derived category of A , and is denoted by D(A ) (resp. D+ (A ), D− (A ), Db (A ), + − b DA 0 (A ), DA 0 (A ), DA 0 (A ), DA 0 (A )).

We have a commutative diagram b b DA 0 (A ) → D (A ) ↓ ↓ ± ± DA 0 (A ) → D (A ) ↓ ↓ DA 0 (A ) → D(A ),

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where all arrows are fully faithful exact functors. We have a canonical exact functor D(A 0 ) → DA 0 (A ) which in general is neither fully faithful nor essentially surjective. Proposition 6.2.4. Let A be an abelian category. For any object A in A , denote also by A the complex ··· → 0 → A → 0 → ···

whose 0-th component is A and whose other components are 0. The functor A → D(A ),

A 7→ A

defines an equivalence between the category A and the full subcategory of D(A ) consisting of complexes X · satisfying H i (X · ) = 0 for all i 6= 0. Proposition 6.2.5. Let A be an abelian category, and let u

v

0 → X· → Y · → Z· → 0

be a short exact sequence of complexes of objects in A . (i) There are canonical morphisms δ i : H i (Z · ) → H i+1 (X · ) such that we have a long exact sequence u

v

δ

· · · → H i (X · ) → H i (Y · ) → H i (Z · ) → H i+1 (X · ) → · · · .

(ii) There exists a morphism w : Z · → X · [1] in D(A ) such that u

is a distinguished triangle.

v

w

X· → Y · → Z· →

Proof. (i) is well-known. Let us prove (ii). We have a distinguished triangle v0

u

w0

X · → Y · → C(u)· →

in K(A ). Since vu = 0, there exists a morphism f : C(u)· → Z · in K(A ) such that v = f v 0 by 6.1.1 (ii). One can check that the following diagram commutes: u

v0

w0

u

v

−δ

· · · → H i (X · ) → H i (Y · ) → H i (C(u)· ) → H i+1 (X · ) → · · · k k ↓f k

· · · → H i (X · ) → H i (Y · ) → H i (Z · ) → H i+1 (X · ) → · · · .

By (i), 6.1.2, and the five lemma, f : C(u)· → Z · is a quasi-isomorphism. So f induces an isomorphism in D(A ). Define w = w0 ◦ f −1 . Then u

v

w

X· → Y · → Z· →

is a distinguished triangle in D(A ).

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Lemma 6.2.6. Let f : Z · → I · be a morphism of complexes in an abelian category A . Suppose that Z · is acyclic, and I · is a bounded below complex of injective objects in A . Then f is homotopic to 0. Proof.

We leave the proof to the reader. Confer [Fu (2006)] 2.1.1 (ii).

Lemma 6.2.7. Let s : I · → X · be a quasi-isomorphism of complexes in an abelian category A . Suppose I · is a bounded below complex of injective objects. Then there exists a quasi-isomorphism t : X · → I · such that ts is homotopic to idI · . Proof.

Choose a distinguished triangle s

C· → I· → X· → . Since s is a quasi-isomorphism, C · is acyclic. By 6.2.6, C · → I · is homotopic to 0. So we have a commutative diagram C· → I· ↓ ↓ idI · 0 → I· in K(A ). By (TR3), there exists t : X · → I · such that (0, idI · , t) is a morphism of triangles. s

→ X· → C· → I· ↓ ↓ idI · ↓t 0 → I·

idI ·

→ I· → .

Then ts is homotopic to idI · , and t is a quasi-isomorphism.

Lemma 6.2.8. Let A be an abelian category, X · , Y · ∈ ob K(A ), I · a bounded below complex of injective objects in A , and s : Y · → I · a quasiisomorphism. Then we have HomK(A ) (X · , I · ) ∼ = HomD(A ) (X · , Y · ). Proof. Let Y ·+ be the category so that objects in Y ·+ are quasis1 isomorphisms Y · → Y 0· , and morphisms in Y ·+ from an object Y · → Y10· s 2 to an object Y · → Y20· are morphisms f : Y10· → Y20· in K(A ) with the s property f s1 = s2 . We claim that Y · → I · is a final object in Y ·+ and hence we have HomD(A ) (X · , Y · ) ∼ =

Y

lim −→

0· ∈ob Y ·+

HomK(A ) (X · , Y 0· ) ∼ = HomK(A ) (X · , I · ).

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Given an arbitrary object Y · → Y 0· in Y ·+ , by 6.2.3, we can find a commutative diagram s

Y · → I· s0 ↓ ↓ s00 f

Y 0· → J ·

in K(A ) such that s00 is a quasi-isomorphism. By 6.2.7, there exists a morphism t : J · → I · such that ts00 = idI · in K(A ). Then tf is a morphism s0

s

in Y ·+ from Y · → Y 0· to Y · → I · . Suppose f1 , f2 : Y 0· → I · are two 0

s

s

morphisms in Y ·+ from Y · → Y 0· to Y · → I · . We then have f 1 s0 = s = f 2 s0 .

By 6.2.3, there exists a quasi-isomorphism s000 : I · → J · such that s000 f1 = s000 f2 . By 6.2.7, there exists a morphism t0 : J · → I · such that t0 s000 = idI · in K(A ). Then we have f1 = t0 s000 f1 = t0 s000 f2 = f2 . s0

s

So there exists one and only one morphism in Y ·+ from Y · → Y 0· to Y · → I ·. Lemma 6.2.9. Let I be a family of objects in an abelian category A such that every object in A can be embedded into an object in I . Then for any X · ∈ ob K(A ) such that H i (X) = 0 for i 0, there exists a quasiisomorphism X · → I · such that I · is a bounded below complex of objects in I. Proof. Define I i = 0 for i 0. Suppose that we have defined I i and morphisms si : X i → I i for all i < n such that H i (X · ) ∼ = H i (I · ) for all n−2 n−1 · n−1 i < n − 1 and such that H (X ) → I /im dI · is injective. Consider the diagram dn−1

i

→ ker dnX · → X n X n−1 /im dn−2 X· n−1 s ↓ ↓ ↓ α

→ I n−1 /im dn−2 I·

A

β

→ B,

where on the first line the morphisms are uniquely determined by the property that i is the inclusion and i ◦ dn−1 is the morphism X n−1 /im dn−2 X· → X n induced by dn−1 , and on the second line, we have · X A = coker(X n−1 /im dn−2 X· B = coker(X n−1 /im dn−2 X·

(sn−1 ,−dn−1 )

→

I n−1 /im dn−2 ⊕ ker dnX · ), I·

→

I n−1 /im dn−2 ⊕ X n ), I·

) (sn−1 ,−dn−1 X·

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and all arrows in the diagram are the canonical morphisms. The first vertical arrow induces an epimorphism H n−1 (X · ) → ker α. Combined with the hypothesis that H n−1 (X · ) → I n−1 /im dn−2 is injective, we see that I· it induces an isomorphism H n−1 (X · ) ∼ ker α. The middle vertical arrow = n · ∼ induces an isomorphism H (X ) = A/im α, and β is injective. Embed B into an object I n in I . Define dn−1 : I n−1 → I n to be the composite I· α

β

I n−1 → I n−1 /im dn−2 → A → B → I n, I·

and define X n → I n to be the composite

X n → B → I n.

Then we have H i (X · ) ∼ is = H i (I · ) for all i < n and H n (X · ) → I n /im dn−1 I· injective. Corollary 6.2.10. Let A be an abelian category and let I be the full subcategory consisting of injective objects. Then the canonical functor K + (I ) → D+ (A ) is fully faithful. If A has enough injective objects, then K + (I ) → D+ (A ) is an equivalence of categories. Proof.

Let I and J be objects in K + (I ). By 6.2.8, we have ∼ HomD(A ) (I, J). HomK(A ) (I, J) =

So the functor K + (I ) → D+ (A ) is fully faithful. If A has enough injective objects, then by 6.2.9, any object in D+ (A ) is quasi-isomorphic to a complex of injective objects. The above functor is then essentially surjective.

6.3

Derived Functors

([Hartshorne (1966)] I 5, [SGA 4] XVII 1.2, [SGA 4 21 ] C.D. II 2.) Let A and B be abelian categories, A 0 a thick abelian subcategory of A , K ∗ (A ) one of the categories + − b K(A ), K + (A ), K − (A ), K b (A ), KA 0 (A ), KA 0 (A ), KA 0 (A ), KA 0 (A ),

D∗ (A ) the corresponding derived category, and F : K ∗ (A ) → K(B) an exact functor. For any X ∈ ob K ∗ (A ), let F 0 (X) : D(B) → (Sets)

be the contravariant functor from D(B) to the category of sets defined by F 0 (X)(Y ) =

lim −→

X 0 ∈ob X +

HomD(B) (Y, F (X 0 ))

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for any Y ∈ ob D(B), where X + is the category so that objects in X + are quasi-isomorphisms X → X 0 , and morphisms in X + from an object s1 s2 X → X10 to an object X → X20 are morphisms f : X10 → X20 in K ∗ (A ) with the property f s1 = s2 . Given a morphism X1 → X2 in K ∗ (A ), define a morphism of functors F 0 (X1 ) → F 0 (X2 ) as follows: For any object X1 → X10 in X1+ , we can find a commutative diagram X1 → X2 ↓ ↓ X10 → X20

such that X2 → X20 is an object in X2+ . The morphism X10 → X20 induces a map HomD(B) (Y, F (X10 )) → HomD(B) (Y, F (X20 )) for any Y ∈ ob D(B). We can take the limit of these maps and get a map F 0 (X1 )(Y ) → F 0 (X2 )(Y ). We thus get a functor F 0 : K ∗ (A ) → Hom(D(B)◦ , (Sets)),

where Hom(D(B)◦ , (Sets)) denotes the category of contravariant functors from D(B) to the category of sets. One can check that F 0 transforms quasiisomorphisms in K ∗ (A ) to isomorphisms in Hom(D(B)◦ , (Sets)). So there exists a unique functor RF : D∗ (A ) → Hom(D(B)◦ , (Sets)),

such that F 0 = RF ◦ Q, where Q : K ∗ (A ) → D∗ (A ) is the canonical functor. We call RF the right derived functor of F . We say that RF is defined at X ∈ ob D∗ (A ) if the functor RF (X) is representable, that is, F 0 (X) ∼ = HomD(B) (−, Z)

for some object Z ∈ D(B). Suppose that RF is defined everywhere on D∗ (A ). Then there exists a functor D∗ (A ) → D(B) that maps each X ∈ ob D∗ (A ) to an object in D(B) representing F 0 (X). We also call this functor the right derived functor of F and denote it by RF : D∗ (A ) → D(B). It it uniquely determined up to isomorphism. For any integer i, we define Ri F (X) = H i (RF (X)). Similarly, for any X ∈ ob K ∗ (A ), let 0

F (X) : D(B) → (Sets)

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be the covariant functor from D(B) to the category of sets defined by 0

F (X)(Y ) =

lim −→

HomD(B) (F (X 0 ), Y )

X 0 ∈ob (X − )◦

for any Y ∈ ob D(B), where X − is the category such that objects in X − are quasi-isomorphisms X 0 → X, and morphisms in X − from an object s1 s2 X10 → X to an object X20 → X are morphisms f : X10 → X20 in K ∗ (A ) with the property s2 f = s1 . It defines a contravariant functor 0

F : K ∗ (A ) → Hom(D(B), (Sets)),

where Hom(D(B), (Sets)) denotes the category of covariant functors from D(B) to the category of sets. One can check that 0 F transforms quasiisomorphisms in K ∗ (A ) to isomorphisms in Hom(D(B), (Sets)). So there exists a unique functor LF : D∗ (A ) → Hom(D(B), (Sets)),

such that 0 F = LF ◦ Q. We call LF the left derived functor of F . We say that LF is defined at X ∈ ob D∗ (A ) if the functor LF (X) is representable, that is, 0

F (X) ∼ = HomD(B) (Z, −)

for some object Z ∈ D(B). Suppose that LF is defined everywhere on D∗ (A ). Then there exists a functor D∗ (A ) → D(B) that maps each X ∈ ob D∗ (A ) to an object in D(B) representing 0 F (X). We also call this functor the left derived functor of F and denote it by LF : D∗ (A ) → D(B). It is uniquely determined up to isomorphism. For any integer i, we define Li F (X) = H i (LF (X)). We leave it for the reader to define left derived functors and right derived functors for contravariant functors. In the following, we state results only for right derived functors of covariant functors. We leave it for the reader to extend them to other derived functors. Proposition 6.3.1. Notation as above. Suppose that there exists a family of objects L of K ∗ (A ) satisfying the following conditions: (a) Any object X in K ∗ (A ) admits a quasi-isomorphism X → X 0 such that X 0 is an object in L . (b) The exact functor F : K ∗ (A ) → K(B) transforms quasiisomorphisms between objects in L to quasi-isomorphisms in K(B).

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(c) For any morphism X 0 → X 00 in K ∗ (A ) such that X 0 and X 00 are objects in L , there exists a distinguished triangle X 0 → X 00 → X 000 → in K ∗ (A ) such that X 000 is also an object in L . Then RF is everywhere defined and RF : D∗ (A ) → D(B) is an exact functor. For any X ∈ ob D∗ (A ), let X → X 0 be a quasi-isomorphism such that X 0 is an object in L . We have RF (X) ∼ = F (X 0 ). Proof. Let X → X 0 be a quasi-isomorphism such that X 0 is an object in L . For any object X → X 00 in X + , we can find a commutative diagram X → X0 ↓ ↓ X 00 → X 000

such that all arrows are quasi-isomorphisms. By condition (a), we may assume that X 000 is an object in L . By condition (b), X 0 → X 000 induces an isomorphism ∼ =

F (X 0 ) → F (X 000 ) in D(B). So we have a one-to-one correspondence HomD(B) (Y, F (X 0 )) ∼ = HomD(B) (Y, F (X 000 )) for any Y ∈ ob D(B). It follows that F 0 (X)(Y ) =

lim −→ 00

X ∈ob X +

HomD(B) (Y, F (X 00 )) ∼ = HomD(B) (Y, F (X 0 )).

0

So the functor F (X) is represented by F (X 0 ). It follows that RF is defined at X and RF (X) ∼ = F (X 0 ). To prove that RF is an exact functor, we use the condition that F is exact, and that any distinguished triangle in D∗ (A ) is isomorphic to a distinguished triangle in K ∗ (A ) whose vertices are objects in L . Corollary 6.3.2. Let K ? (A ) be one of the categories + − b K(A ), K + (A ), K − (A ), K b (A ), KA 0 (A ), KA 0 (A ), KA 0 (A ), KA 0 (A )

contained in K ∗ (A ) and let D? (A ) be the corresponding derived category. Suppose that the conditions of 6.3.1 hold, and suppose furthermore that any object X in K ? (A ) admits a quasi-isomorphism from X to an object in L ∩ ob K ? (A ). Then R(F |K ? (A ) ) is defined everywhere on D? (A ) and R(F |K ? (A ) ) ∼ = (RF )|D? (A ) .

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Corollary 6.3.3. Let A , B and C be abelian categories, A 0 (resp. B 0 ) a thick abelian subcategory of A (resp. B), K ∗ (A ) (resp. K ? (B)) one of the categories + − b K(A ), K + (A ), K − (A ), K b (A ), KA 0 (A ), KA 0 (A ), KA 0 (A ), KA 0 (A ) + − b (resp. K(B), K + (B), K − (B), K b (B), KB0 (B), KB 0 (B), KB0 (B), KB0 (B)),

D∗ (A ) (resp. D? (B)) the corresponding derived category, F : K ∗ (A ) → K ? (B) and G : K ? (B) → K(C ) exact functors. Suppose that the following conditions hold: (a) There exists a family L (resp. M ) of objects in K ∗ (A ) (resp. ? K (B)) such that any object X in K ∗ (A ) (resp. K ? (B)) admits a quasiisomorphism from X to an object X 0 in L (resp. M ), and for any morphism X 0 → X 00 in K ∗ (A ) (resp. K ? (B)) such that X 0 and X 00 are objects in L (resp. M ), there exists a distinguished triangle X 0 → X 00 → X 000 →

in K ∗ (A ) (resp. K ? (B)) such that X 000 is also an object in L (resp. M ). (b) F (resp. G) transforms quasi-isomorphisms between objects in L (resp. M ) to quasi-isomorphisms in K ? (B) (resp. K(C )). (c) F (L ) ⊂ M . Then RF , RG and R(GF ) are everywhere defined, and R(GF ) ∼ = RG◦RF . Proposition 6.3.4. Let A and B be abelian categories. Suppose that A has enough injective objects. (i) For any exact functor F : K + (A ) → K(B), RF is everywhere defined on D+ (A ). For any object X in K + (A ), we can find a quasiisomorphism X → I such that I is a bounded below complex of injective objects in A , and we have RF (X) ∼ = F (I). (ii) Let F : A → B be a left exact functor. Denote the functor K(A ) → K(B) induced by F also by F . By (i), RF is everywhere defined on D+ (A ). Suppose that RF has finite cohomological dimension. Then RF is everywhere defined on D(A ). Here we say RF has finite cohomological dimension if there exists a nonnegative integer n such that Ri F (X) = 0 for any i > n and any X ∈ ob A . Proof. (i) Let L be the family of objects in K + (A ) consisting of bounded below complexes of injective objects in A . Since A has enough injective objects, by 6.2.9, for any object X in K + (A ), we can find a quasi-isomorphism

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X → I so that I is an object in L . If s : I1 → I2 is a quasi-isomorphism of objects in L , then by 6.2.7, s is an isomorphism in K + (A ). Hence F (s) is an isomorphism in K(B). Our assertion then follows from 6.3.1. (ii) An object X in A is called RF -acyclic if Ri F (X) = 0 for all i > 0. Injective objects are RF -acyclic. If 0 → X 0 → X → X 00 → 0

is an exact sequence of objects in A , and X 0 and X 00 are RF -acyclic, then X is also RF -acyclic, and the sequence 0 → F (X 0 ) → F (X) → F (X 00 ) → 0

is exact. Take L to be the family of objects in K(A ) consisting of complexes of RF -acyclic objects. Let n be an integer such that Ri F (X) = 0 for any i > n and any X ∈ ob A , and let d0

d1

0 → I0 → I1 → · · ·

be an injective resolution of an object X in A . Then dn

dn+1

0 → I n → I n+1 → · · ·

is an injective resolution of ker dn . For any i > 0, we have ∼ H i (F (I · [n])) ∼ Ri F (ker dn ) = = Ri+n F (X) = 0. So ker dn is RF -acyclic. Hence

0 → I 0 → · · · → I n−1 → ker dnI· → 0

is an RF -acyclic resolution of X. Let X · be a complex in K(A ), and let I ·· be the Cartan–Eilenberg resolution of X · . For each j, 0 → I 0j → I 1j → · · ·

is an injective resolution of X j . We have a quasi-isomorphism from X · to the complex associated to the truncated bicomplex 0 0 ↑ ↑ n,j · · · → ker d1 → ker dn,j+1 → ··· 1 ↑ ↑ · · · → I n−1,j → I n−1,j+1 → · · · ↑ ↑ .. .. . . ··· →

I

↑

0,j

↑ 0

→

I

↑

0,j+1

↑ 0

→ ···

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and this is a complex of RF -acyclic objects. Hence every object X in K(A ) admits a quasi-isomorphism from X to an object in L . Next we show that F transforms quasi-isomorphisms of objects in L into quasi-isomorphisms in K(B). Then (ii) follows from 6.3.1. It suffices to show that F transforms acyclic complexes in L to acyclic complexes. Let X · be an acyclic complex of RF -acyclic objects, and let Z i = ker diX · . We have short exact sequences 0 → Z i−1 → X i−1 → Z i → 0. Since X i are RF -acyclic, for any q > 0, we have Rq F (Z i ) ∼ = Rq+1 F (Z i−1 ) ∼ = ··· ∼ = Rq+n F (Z i−n ) = 0. So Z i are RF -acyclic. It follows that 0 → F (Z i−1 ) → F (X i−1 ) → F (Z i ) → 0 are exact, and hence F (X · ) is acyclic.

Proposition 6.3.5. Let A and B be abelian categories and let F : A → B be a left exact functor. Suppose that A has enough injective objects. Then for any K · ∈ ob D+ (A ), we have two biregular spectral sequences E2pq = Rp F (H q (K · )) ⇒ Rp+q F (K · ),

E1pq = Rq F (K p ) ⇒ Rp+q F (K · ).

Suppose furthermore that RF has finite cohomological dimension. Then we have the same result for any K · ∈ ob D(A ). Proof. Let us prove the second part of the proposition. Let n be an integer such that Ri F (X) = 0 for any object X in A and any i > n. Given an object X · in K(A ), let Z j = ker djX · ,

B j = im dj−1 X· ,

H j = Z j /B j .

Choose a Cartan–Eilenberg resolution I ·· of X · . Then for each j, ·,j ·,j ·,j−1 ·,j ·,j−1 ·,j (I ·,j , d·,j , d1 ), (ker d·,j , d1 ) 1 ), (ker d2 , d1 ), (im d2 2 /im d2

are injective resolutions of X j , Z j , B j , H j , respectively. For any complex Y · , let τ≤n Y · be the truncated complex · · · → Y n−1 → ker dnY · → 0. Then ·,j ·,j ·,j−1 (τ≤n (I ·,j ), d·,j ), d·,j 1 ), (τ≤n (ker d2 ), d1 ), (τ≤n (im d2 1 ), ·,j−1 (τ≤n (ker d·,j ), d·,j 2 /im d2 1 )

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are resolutions by RF -acyclic objects of X j , Z j , B j , H j , respectively. Moreover, we have short exact sequences of complexes ·,j ·,j−1 0 → τ≤n (im d·,j−1 ) → τ≤n (ker d·,j ) → 0, 2 2 ) → τ≤n (ker d2 /im d2

·,j ·,j 0 → τ≤n (ker d·,j 2 ) → τ≤n (I ) → τ≤n (im d2 ) → 0.

Let I 0·· be the truncated bicomplex 0 0 ↑ ↑ n,j+1 · · · → ker dn,j → ··· 1 → ker d1 ↑ ↑ · · · → I n−1,j → I n−1,j+1 → · · · ↑ ↑ .. .. . . ··· →

↑ I 0,j ↑ 0

→

↑ I 0,j+1 ↑ 0

→ ···

and denote objects associated to this complex by putting the supscript 0 . Then the complexes 0·,j 0·,j 0·,j−1 0·,j 0·,j−1 0·,j (I 0·,j , d0·,j , d1 ), (ker d0·,j , d1 ) 1 ), (ker d2 , d1 ), (im d2 2 /im d2

can be identified with ·,j ·,j ·,j−1 (τ≤n (I ·,j ), d·,j ), d·,j 1 ), (τ≤n (ker d2 ), d1 ), (τ≤n (im d2 1 ), ·,j−1 (τ≤n (ker d·,j ), d·,j 2 /im d2 1 )

respectively, and hence are resolutions by RF -acyclic objects of X j , Z j , B j , H j , respectively. The two spectral sequences are those associated to the bicomplex F (I 0·· ). Let X be a scheme and let A be the abelian category of sheaves of abelian groups on X. Denote the derived category D(A ) by D(X). Let A 0 be the thick abelian subcategory consisting of torsion sheaves (resp. constructible sheaves). Denote the derived category DA 0 (A ) by Dtor (X) (resp. Dc (X)). Let A be a ring, let A be the abelian category of sheaves of A-modules on X, and let A 0 be the thick abelian subcategory consisting of constructible sheaves of A-modules. Denote the derived category D(A ) (resp. DA 0 (A )) by D(X, A) (resp. Dc (X, A)). We get the full subcate∗ gories D∗ (X), Dtor (X), Dc∗ (X), D∗ (X, A), Dc∗ (X, A) by taking ∗ = +, −, b.

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The category of sheaves of abelian groups has enough injective objects. We can define the right derived functor RΓ(X, −) : D+ (X) → D+ ((Ab. gps)) of the functor Γ(X, −) of taking global sections, where (Ab. gps) is the category of abelian groups. If RΓ(X, −) has finite cohomological dimension, we can extend RΓ(X, −) to RΓ(X, −) : D(X) → D((Ab. gps)). For any sheaf of abelian groups F on X and any i, we have H i (RΓ(X, F )) = H i (X, F ). Let f : X → Y be a morphism of schemes. We can define the right derived functor Rf∗ : D+ (X) → D+ (Y ) of f∗ . If Rf∗ has finite cohomological dimension, we can then define Rf∗ : D(X) → D(Y ). For any sheaf of abelian groups F on X and any i, we have H i (Rf∗ F ) = Ri f∗ F . 6.4

RHom(−, −) and − ⊗L A −

([Hartshorne (1966)] I 6, [SGA 4] XVII 1.1, [SGA 4 21 ] C.D. II 3.) Let A be an abelian category. For all complexes X · and Y · of objects in A , define a complex Hom· (X · , Y · ) of abelian groups as follows: Take Y Homn (X · , Y · ) = HomA (X i , Y i+n ). i

For any

f = (fi ) ∈ Homn (X · , Y · ) = define

Y i

df = ((df )i ) ∈ Homn+1 (X · , Y · ) = by

HomA (X i , Y i+n ),

Y

HomA (X i , Y i+n+1 )

i

n i (df )i = di+n Y · ◦ fi − (−1) fi+1 ◦ dX · .

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We have dd = 0. Note that the kernel of d : Homn (X · , Y · ) → Homn+1 (X · , Y · ) can be identified with the group of morphisms of complexes from X · to Y · [n], and the image of d : Homn−1 (X · , Y · ) → Homn (X · , Y · ) can be identified with the subgroup of morphisms of complexes from X · to Y · [n] homotopic to 0. So we have H n (Hom· (X · , Y · )) ∼ = HomK(A ) (X · , Y · [n]). Define ∼ =

Hom· (X · , Y · [1]) → Hom· (X · , Y · )[1] so that it is the identity through the identification Y Homn (X · , Y · [1]) = HomA (X i , Y i+n+1 ) = (Hom· (X · , Y · )[1])n . i

Define

∼ =

Hom· (X · [−1], Y · ) → Hom· (X · , Y · )[1] so that it is (−1)n+1 id through the identification Y Homn (X · [−1], Y · ) = HomA (X i , Y i+n+1 ) = (Hom· (X · , Y · )[1])n . i

One can check they are morphisms of complexes. Moreover, the following diagram is anti-commutative: ∼ =

Hom· (X · [−1], Y · [1]) → Hom· (X · [−1], Y · )[1]. ∼ ↓∼ = = ↓ ∼ =

Hom· (X · , Y · [1])[1] → Hom· (X · , Y · )[1][1].

Fix an object X · (resp. Y · ) in K(A ). Then Hom· (X · , −) (resp. Hom· (−, Y · )) transforms distinguished triangles to distinguished triangles, and hence Hom· (−, −) : K(A ) × K(A ) → K((Ab.gps)) is an exact functor contravariant in the first variable, and covariant in the second variable. As an example, given a distinguished triangle u

v

w

X1· → X2· → C(u) →,

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where C(u) is the mapping cone of u, let us verify that v∗

u∗

w0

Hom· (C(u), Y · ) → Hom· (X2· , Y · ) → Hom· (X1· , Y · ) →

is a distinguished triangle, where w0 is defined by the commutative diagram Hom· (X1· , Y · ) k

Hom· (X1· , Y · )

w0

→ (w[−1])∗

→

Hom· (C(u), Y · )[1] ↑

Hom· (C(u)[−1], Y · ).

It suffices to show that u∗

−v ∗ [1]

w0

Hom· (X2· , Y · ) → Hom· (X1· , Y · ) → Hom· (C(u), Y · )[1] → is a distinguished triangle. We need to define an isomorphism Hom· (C(u), Y · )[1] ∼ = C(u∗ ) so that the following diagram commutes: −v ∗ [1]

w0

Hom· (X1· , Y · ) → Hom· (C(u), Y · )[1] → Hom· (X2· , Y · )[1] k ↓∼ k = · · · ∗ · C(u ) → Hom (X2· , Y · )[1]. Hom (X1 , Y ) → For each integer n, the isomorphism (Hom· (C(u), Y · )[1])n → (C(u∗ ))n is defined to be the composite of the following isomorphisms:

=

(Hom· (C(u), Y · )[1])n Y Hom(C(u)i , Y i+n+1 ) i

=

Y i

∼ = ((−1)n+1 id,−id)

→ = =

Y

Hom(X1i+1 ⊕ X2i , Y i+n+1 ) Hom(X1i+1 , Y i+n+1 )

i

Y

MY

Hom(X2i , Y i+n+1 )

i

Hom(X1i+1 , Y i+n+1 )

i

n

Hom (X1· , Y · ) ∗ n (C(u )) .

M

MY

Hom(X2i , Y i+n+1 )

i

n+1

Hom

(X2· , Y · )

Lemma 6.4.1. Let X · be an acyclic complex of objects in A and let Y · be a bounded below complex of injective objects in A . Then Hom· (X · , Y · ) is acyclic.

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Proof.

We have H n (Hom· (X · , Y · )) ∼ = HomK(A ) (X · , Y · [n]).

By 6.2.6, we have HomK(A ) (X · , Y · [n]) = 0. Our assertion follows.

Suppose that A is an abelian category with enough injective objects. For any fixed X · ∈ ob K(A ), the right derived functor of Hom· (X · , −) : K + (A ) → K((Ab. gps))

is defined everywhere on D+ (A ) by 6.3.4 (i). Denote it by RII Hom· (X · , −) : D+ (A ) → D((Ab. gps)). We thus get a functor RII Hom· (−, −) : K(A ) × D+ (A ) → D((Ab. gps)).

By 6.4.1, the functor RII Hom· (−, Y ) transforms quasi-isomorphisms in K(A ) to isomorphisms in D((Ab. gps)). So it factors through D(A ) × D+ (A ) and defines a functor D(A ) × D+ (A ) → D((Ab. gps)),

which we denote by RI RII Hom· (−, −), or simply RHom(−, −). Define Exti (X · , Y · ) = H i (RHom· (X · , Y · ))

for any X ∈ ob D(A ) and Y ∈ ob D+ (A ). Proposition 6.4.2. Let A be an abelian category with enough injective objects. For any X ∈ ob D(A ) and Y ∈ ob D+ (A ), we have Exti (X · , Y · ) ∼ = HomD(A ) (X · , Y · [i]).

Proof. Let Y · → I · be a quasi-isomorphism so that I · is a bounded below complex of injective objects in A . We have Exti (X ·, Y · ) = H i (RHom· (X ·, Y · )) ∼ = H i (Hom· (X ·, I · )) ∼ = HomK(A ) (X ·, I · [i]). Our assertion follows from 6.2.8.

For any X · , Y · ∈ ob D(A ), define

Exti (X · , Y · ) ∼ = HomD(A ) (X · , Y · [i]).

By 6.4.2, this coincides with our previous definition of Exti (X · , Y · ) if A has enough injective objects, X ∈ ob D(A ) and Y ∈ ob D+ (A ). By 6.1.1 (ii) and 6.2.5 (ii), we have the following.

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Proposition 6.4.3. Let A be an abelian category and let 0 → X· → Y · → Z· → 0 be an exact sequence of complexes of objects in A . For any W · ∈ ob D(A ), we have the following exact sequences · · · → Exti (W ·, X · ) → Exti (W ·, Y · ) → Exti (W ·, Z · ) → Exti+1 (W ·, X · ) → · · · ,

· · · ← Exti (X ·, W · ) ← Exti (Y ·, W · ) ← Exti (Z ·, W · ) ← Exti−1 (X ·, W · ) ← · · · . Let X be a scheme and let A be a ring. Taking A to be the abelian category of sheaves of A-modules on X, we get the functors RHom : D(X, A) × D+ (X, A) → D(Ab gps) and ExtiA (−, −) : D(X, A) × D(X, A) → (Ab. gps). Given two complexes of sheaves of A-modules F · and G · , we can define a complex of sheaves of A-modules H om· (F · , G · ) in the same way as above by replacing Hom by the sheafified H om. We can then define the functors RH om : D(X, A) × D+ (X, A) → D(X, A) and E xtiA (−, −) = H i (RH om(−, −)). Given two complexes F · and G · of sheaves of A-modules on X, define a complex F · ⊗A G · as follows: Take M (F · ⊗A G · )n = F i ⊗A G n−i . i

Define

d : (F · ⊗A G · )n → (F · ⊗A G · )n+1 by d|F i ⊗A G n−i = diF · ⊗ idG n−i + (−1)i idF i ⊗ dn−i G· . Define (F · [1]) ⊗A G · → (F · ⊗A G · )[1] so that its restriction to F i ⊗A G n−i+1 is the identity, and define F · ⊗A (G · [1]) → (F · ⊗A G · )[1]

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so that its restriction to F i ⊗A G n−i+1 is (−1)i id. One checks that the diagram (F · [1]) ⊗A (G · [1]) → (F · ⊗A (G · [1]))[1] ↓ ↓ ((F · [1]) ⊗A G · )[1] → (F · ⊗A G · )[1][1]

is anti-commutative, and

− ⊗A − : K(X, A) × K(X, A) → K(X, A) is an exact functor, where K(X, A) is the triangulated category of complexes of sheaves of A-modules on X. We have an isomorphism of complexes F · ⊗A G · → G · ⊗A F · defined by xi ⊗ y j 7→ (−1)ij y j ⊗ xi

for any xi ∈ F i and y j ∈ G j .

Lemma 6.4.4. Let X be a scheme, A a ring, F · and G · complexes of sheaves of A-modules on X. Suppose that the following conditions hold: (a) All components of G are flat sheaves of A-modules. (b) Either F · or G · is acyclic. (c) Either both F · and G · are bounded above, or G · is bounded. Then F · ⊗A G · is acyclic. Proof.

By the condition (c), the two spectral sequences 0 00

q E2pq = HIp HII (F · ⊗A G · ) ⇒ H p+q (F · ⊗A G · ), p E2pq = HII HIq (F · ⊗A G · ) ⇒ H p+q (F · ⊗A G · )

associated to the bicomplex F · ⊗A G · are biregular. If F · is acyclic, then by condition (a), we have HIq (F · ⊗A G · ) = 0 for all q. So 00 E2pq = 0 for all p and q, and hence H n (F · ⊗A G · ) = 0 for all n. Suppose G · is acyclic. By condition (c), G · is bounded above, say of the form dm−2

dm−1

· · · → G m−1 → G m → 0. Then we have short exact sequences 0 → ker dm−1 → G m−1 → G m → 0, G· m−2 0 → ker dG · → G m−2 → ker dm−1 → 0, G· .. .

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Since G q are flat, it follows from the above exact sequences that ker dqG · are all flat. Hence we have short exact sequences 0 → F p ⊗A ker dm−1 → F p ⊗A G m−1 → F p ⊗A G m → 0, G· m−2 p → 0, 0 → F ⊗A ker dG · → F p ⊗A G m−2 → F p ⊗A ker dm−1 G· .. . It follows that · · · → F p ⊗A G q → F p ⊗A G q+1 → · · ·

are exact. So 0 E2pq = 0 for all p and q, and hence H n (F · ⊗A G · ) = 0 for all n. Let L be the family of objects in K − (X, A) consisting of bounded above complexes of flat sheaves A-modules. For each fixed object F · in K − (X, A), by 6.4.4, the functor F · ⊗A − transforms quasi-isomorphisms in L to quasi-isomorphisms. On the other hand, any sheaf of A-modules on X is a quotient of a flat sheaf. Indeed, any sheaf of A-modules is a L fU quotient of a sheaf of the form U AU , where U → X are etale X-schemes and AU = fU! A. By the dual version of 6.2.9, for any G · ∈ ob K − (X, A), there exists a quasi-isomorphism from an object in L to G · . (Confer 6.4.5 below.) By the dual version of 6.3.1, the left derived functor of F · ⊗A − is defined everywhere on D− (X, A). Denote the derived functor by LII (F · ⊗A −) : D− (X, A) → D− (X, A). So we get a functor LII (− ⊗A −) : K − (X, A) × D− (X, A) → D− (X, A). By 6.4.4, the functor LII (− ⊗A G · ) transforms quasi-isomorphisms in K − (X, A) to isomorphisms in D− (X, A). So it factors through D− (X, A) and defines a functor D− (X, A) × D− (X, A) → D− (X, A), which we denote by LI LII (−⊗A −). Similarly we can define LII LI (−⊗A −), and we have LII LI (− ⊗A −) ∼ = LI LII (− ⊗A −). · · − We denote LI LII (− ⊗A −) by − ⊗L A −. For any F , G ∈ ob D (X, A), we define · Tori (F · , G · ) = H −i (F · ⊗L A G ).

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Lemma 6.4.5. Let X be a noetherian scheme, A a noetherian ring, and F · a complex of sheaves of A-modules on X such that H i (F · ) are constructible sheaves of A-modules for all i and H i (F · ) = 0 for i 0. Then there exists a quasi-isomorphism F 0· → F · such that F 0· is a bounded above complex of flat constructible sheaves of A-modules. Proof. We use the dual argument of the proof of 6.2.9. Define F 0i = 0 for i 0. Suppose that we have defined constructible flat sheaves of A-modules F 0i and morphisms F 0i → F i for all i > n such that H i (F 0· ) ∼ = H i (F · ) n+1 for all i > n+1 and such that ker dn+1 (F · ) is surjective. Consider F 0· → H the diagram α

A → ↓

B ↓

p

β

→ ker dn+1 F 0· ↓ dn

n+1 F n → F n /im dn−1 F · → ker dF · ,

where on the second line the morphisms are uniquely determined by the property that p is the projection and dn ◦ p is the morphism F n → ker dn+1 F· induced by dnF · , and on the first line, we have A = F n ×ker dn+1 ker dn+1 F 0· , · F

B = F n /im dn−1 ker dn+1 F · ×ker dn+1 F 0· , · F

and all arrows in the diagram are the canonical morphisms. The last vertical arrow induces a monomorphism coker β → H n+1 (F · ). Combined with the n+1 hypothesis that ker dn+1 (F · ) is surjective, we see that it induces F 0· → H n+1 ∼ an isomorphism coker β = H (F · ). The middle vertical arrow induces n · ∼ an isomorphism ker β = H (F ). So we have an exact sequence β

n+1 0 → H n (F · ) → B → ker dn+1 (F · ) → 0. F 0· → H

Since H n (F · ), H n+1 (F · ) and ker dn+1 F 0· are constructible, B is also constructible. We will construct a constructible flat sheaf of A-modules F 0n and a morphism γ : F 0n → A such that αγ is surjective. Define dnF 0· : F 0n → F 0n+1 to be the composite γ

β

α

0n+1 F 0n → A → B → ker dn+1 , F 0· → F

and define F 0n → F n to be the composite α

F 0n → A → F n .

Then H i (F 0· ) ∼ = H i (F · ) for all i > n and ker dnF 0· → H n (F · ) is surL jective. To construct F 0n , choose an epimorphism U AU → A , where

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fU

U → X are some etale X-schemes and AU = fU! A. Since α is necessarily L surjective, the composite U AU → A → B is surjective. Since B is conLk structible, there exists a finite family U1 , . . . , Uk such that i=1 AUi → B L k is surjective. We then take F 0n = i=1 AUi .

Let F · be a bounded complex of sheaves of A-modules on X. We say that F · has finite Tor-dimension if there exists an integer n such that Tori (F · , M ) = 0 for any i > n and any constant sheaf of A-modules M on X. Proposition 6.4.6. Let X be a noetherian scheme, A a noetherian ring, and F · a bounded complex of sheaves of A-modules. The following conditions are equivalent: (i) There exists a quasi-isomorphism F 0· → F · such that F 0· is a bounded complex of constructible flat sheaves of A-modules. (ii) F · has finite Tor-dimension and H i (F ) are constructible for all i. Proof. (i)⇒(ii) is clear. Suppose (ii) holds. Let n be an integer such that Tori (F · , M ) = 0 for any i > n and any constant sheaf of A-modules M on X. Taking M = A, we get H i (F · ) = 0 for all i < −n. Let F 0· → F be the quasi-isomorphism constructed in 6.4.5. Then · · · → F 0−n−1 → F 0−n → 0 is a flat resolution of coker d−n−1 F 0· . So we have −i ∼ Tori (coker d−n−1 (F 0· [−n] ⊗A M ) ∼ = H −i−n (F 0· ⊗A M ) F 0· , M ) = H ∼ = Tori+n (F · , M ) = 0

for all i > 0. Hence coker d−n−1 is a flat sheaf of A-modules. It is also F 0· constructible. Let τ≥−n F 0· be the truncated complex 0 → coker d−n−1 → F 0−n+1 → F 0−n+2 → · · · . F 0· It is a complex of constructible flat sheaves of A-modules. Moreover, τ≥−n F 0· → τ≥−n F · is a quasi-isomorphism. Since F · is bounded, we have τ≥−n F · = F · if we take n sufficiently large. b Let X be a scheme and let A be a ring. Denote by Dtf (X, A) the full subb category of D (X, A) consisting of objects with finite Tor-dimension. Then b Dtf (X, A) is a triangulated subcategory of Db (X, A). Suppose that X is a b noetherian scheme and A a noetherian ring. Denote by Dctf (X, A) the full b · subcategory of D (X, A) consisting of objects F with finite Tor-dimension,

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b and such that H i (F ) are constructible for all i. Then Dctf (X, A) is a trib b angulated subcategory of D (X, A). By 6.4.6, any object in Dctf (X, A) is isomorphic to a bounded complex of constructible flat sheaves of A-modules on X. Let L be the family of objects in K b (X, A) consisting of bounded complexes of flat sheaves of A-modules. For each fixed object F · in K(X, A), by 6.4.4, the functor F · ⊗A − transforms quasi-isomorphisms in L to quasi-isomorphisms. On the other hand, using an argument similar to the proof of 6.4.6, one can show that for any G · ∈ ob K b (X, A) with finite Tordimension, there exists a quasi-isomorphism from an object in L to G · . By the dual version of 6.3.1, the left derived functor of F · ⊗A − is defined b everywhere on Dtf (X, A). Denote the derived functor by b (X, A) → D(X, A). LII (F · ⊗A −) : Dtf

So we get a functor b LII (− ⊗A −) : K(X, A) × Dtf (X, A) → D(X, A).

By 6.4.4, the functor LII (− ⊗A G · ) transforms quasi-isomorphisms in K(X, A) to isomorphisms in D(X, A). So it factors through D(X, A) and defines a functor b − ⊗L A − : D(X, A) × Dtf (X, A) → D(X, A).

For any F · , G · ∈ ob K(X, A), the canonical evaluation morphism F · ⊗A H om· (F · , G · ) → G · is not a morphism of complexes, but the evaluation morphism H om· (F · , G · ) ⊗A F · → G · b is a morphism of complexes. If F · ∈ ob Dtf (X, A) and G · ∈ ob D+ (X, A), then the second evaluation morphism induces a morphism · · Ev : RH om(F · , G · ) ⊗L AF →G

in D(X, A). Let F 0· → F · and G · → I · be quasi-isomorphisms such that F 0· is a bounded complex of flat sheaves of A-modules, and I · is a bounded below complex of injective sheaves of A-modules. Ev is the composite · ∼ · · 0· RH om(F · , G · ) ⊗L A F = H om(F , I ) ⊗A F

→ H om(F · , I · ) ⊗A F ·

Ev → I· ∼ = G ·.

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b Proposition 6.4.7. Let X be a scheme, A a ring, F · ∈ ob Dtf (X, A), · + · and G ∈ ob D (X, A). For any H ∈ ob D(X, A) and any morphism · · φ : H · → RH om(F · , G · ), let F (φ) : H · ⊗L A F → G be the composite · H · ⊗L AF

φ⊗L A id

Ev

· · → RH om(F · , G · ) ⊗L AF →G .

Then F defines a one-to-one correspondence ∼ =

· · F : HomD(X,A) (H · , RH om(F · , G · )) → HomD(X,A) (H · ⊗L A F , G ).

Proof. We may assume that F · is a bounded complex of flat sheaves and G · is a bounded below complex of injective sheaves. Then Hom· (F · , G · ) is a bounded below complex of injective sheaves by 5.6.8. We have · · · ∼ 0 · · · HomD(X,A) (H · ⊗L A F , G ) = H (Hom (H ⊗A F , G )) ∼ = H 0 (Hom· (H · , H om(F · , G · )) ∼ HomD(X,A) (H · , RH om(F · , G · )). =

Lemma 6.4.8. Let A be a ring, let di

· · · → K i → K i+1 → · · · be a complex of A-modules, let .. . ↑

di,j+1

.. . ↑

· · · → C i,j+1 1→ C i+1,j+1 → · · · ↑ ↑ di+1,j di,j 2 2 ··· →

··· →

dij

1 C i+1,j C ij → ↑ ↑ .. .. . . ↑ ↑

di0

1 C i0 → C i+1,0 ↑ ↑ 0 0

→ ···

→ ···

be a bicomplex of A-modules, and let f · : (K · , d· ) → (C ·0 , d·0 1 ) be a morphism of complexes. Suppose for each i, the sequence di0

2 C i1 → · · · 0 → K i → C i0 →

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is exact. Let dij

2 Z ij = ker(C ij → C i,j+1 ).

Suppose for each pair (i, j), there exists a left inverse pij : C ij → Z ij of the inclusion Z ij ,→ C ij such that the following diagram commutes: C ij pij ↓ Z ij

dij 1

→

dij 1 |Z ij

→

C i+1,j ↓ pi+1,j Z i+1,j

Then K · is homotopically equivalent to the complex associated to the bicomplex C ·· . L i,j+1 Proof. Let Dij = Z ij Z . Define ij i+1,j dij 1 :D →D

ij i+1,j to be the homomorphisms induced by the restrictions of dij 1 : C →C i,j+1 i,j+1 i+1,j+1 ij i,j+1 and d1 :C →C to Z and Z , respectively, and define ij i,j+1 dij 2 :D →D

by dij 2 (x1 , x2 ) = (x2 , 0) for any (x1 , x2 ) ∈ Z ij morphisms

L

Z i,j+1 . Then D·· is a bicomplex, and the homo-

C ij → Dij ,

x 7→ (pij (x), dij 2 (x))

define an isomorphism of bicomplexes. For each n, let M M Dn = Dij = (Z ij ⊕ Z i,j+1 ) i+j=n,j≥0

i+j=n,j≥0

and define dn : Dn → Dn+1

L L to be i+j=n,j≥0 (dij (−1)i dij 1 2 ). To prove the lemma, it suffices to prove · that the complex K is homotopically equivalent to the complex D· . Note that for each n, the homomorphism f n : K n → C n0 induces an isomorphism of K n with Z n0 . Define φn : K n → Dn

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so that any element x in K n is mapped to M M (f n (x), 0), (0, 0), . . . ∈ Dn = (Z n0 ⊕ Z n1 ) (Z n−1,1 ⊕ Z n−1,2 ) ··· ,

and define

ψn : Dn → K n so that any element (xn0 , y n1 ), (xn−1,1 , y n−1,2 ), . . . M M ∈ Dn = (Z n0 ⊕ Z n1 ) (Z n−1,1 ⊕ Z n−1,2 ) ···

is mapped to the preimage of xn0 ∈ im f n in K n . Then φ· : K · → D· and ψ · : D· → K · are morphisms of complexes, and ψ · ◦ φ· = idK · . Define H n : Dn → Dn−1 so that any element (xn0 , y n1 ), (xn−1,1 , y n−1,2 ), . . . M M ∈ Dn = (Z n0 ⊕ Z n1 ) (Z n−1,1 ⊕ Z n−1,2 ) ···

is mapped to (0, (−1)n−1 xn−1,1 ), (0, (−1)n−2 xn−2,2 ), . . . M M ∈ Dn−1 = (Z n−1,0 ⊕ Z n−1,1 ) (Z n−2,1 ⊕ Z n−2,2 ) ··· . Then we have

dn−1 H n + H n+1 dn = idDn − φn ψ n . So K · is homotopically equivalent to D· .

Corollary 6.4.9. Let X be a scheme, A a ring, F · a complex of sheaves of A-modules on X, C · (F i ) the Godement resolution of F i for any i, τ≤n C · (F i ) the truncated Godement resolution 0 → C 0 (Fi ) → · · · → C n−1 (Fi ) → ker C n (F i ) → C n+1 (F i ) → 0

for any i and any n ≥ 0, and P the set of geometric points of X in the definition of the Godement resolution. Then for any geometric point t ∈ P , Ft· is homotopically equivalent to the complex associated to the bicomplex (C · (F · ))t , and to the complex associated to the bicomplex (τ≤n C · (F · ))t .

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Proof.

This follows from 6.4.8, the fact that the complexes 0 → Fit → (C · (F i ))t ,

0 → Fit → (τ≤n C · (F i ))t are split, and the splitting is functorial with respect to Fi . Confer the proof of 5.6.13 (iii). Corollary 6.4.10. Let X be a scheme, A a ring, F · and G · bounded above complexes of sheaves of A-modules on X, and G 0· → G · a quasiisomorphism such that G · is a bounded above complex of flat sheaves of A-modules on X. Denote the complex associated to the bicomplex C · (G 0· ) (resp. τ≤n C · (F · )) of Godement resolutions (resp. truncated Godement resolutions) by the same notation. Then we have · ∼ · · 0· F · ⊗L A G = F ⊗A C (G ), · ∼ · · 0· F · ⊗L A G = F ⊗A τ≤n C (G ).

Suppose that G · has finite Tor-dimension, and G 0· is a bounded complex of flat sheaves of A-modules quasi-isomorphic to G · . Then the above results hold for any complex F · of sheaves of A-modules. Proof. Let P be the set of geometric points of X in the definition of the Godement resolution. By 6.4.9, for any t ∈ P , the canonical morphisms Gt0· → (C · (G 0· ))t ,

Gt0· → (τ≤n C · (G 0· ))t are homotopically invertible. It follows that (F · ⊗A G 0· )t → (F · ⊗A C · (G 0· ))t ,

(F · ⊗A G 0· )t → (F · ⊗A τ≤n C · (G 0· ))t

are homotopically invertible. Hence F · ⊗A G 0· → F · ⊗A C · (G 0· ),

F · ⊗A G 0· → F · ⊗A τ≤n C · (G 0· )

· are quasi-isomorphisms. We have F · ⊗A G 0· ∼ = F · ⊗L A G . Our assertion follows.

Let A be a ring, let di

· · · → C i → C i+1 → · · ·

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be a complex of A modules, and let Z i = ker di ,

B i = im di−1 ,

H i = Z i /B i .

We have short exact sequences 0 → Z i → C i → B i+1 → 0, 0 → B i → Z i → H i → 0.

We say that the complex C · is split if the above short exact sequences are split for all i. Lemma 6.4.11. Let A be a ring, and let C · be a split complex of A modules. Then C · is homotopically equivalent to the complex 0

· · · → H i (C · ) → H i+1 (C · ) → · · · . Proof.

Notation as above. We have isomorphisms Ci ∼ = Z i ⊕ B i+1 ∼ = H i ⊕ B i ⊕ B i+1 .

Through these isomorphisms, di : C i → C i+1 can be identified with the homomorphisms di : H i ⊕ B i ⊕ B i+1 → H i+1 ⊕ B i+1 ⊕ B i+2 ,

(hi , bi , bi+1 ) 7→ (0, bi+1 , 0).

The homomorphisms φi : H i → H i ⊕ B i ⊕ B i+1 , i

i

i

ψ :H ⊕B ⊕B

i+1

i

→H ,

hi 7→ (hi , 0, 0),

(hi , bi , bi+1 ) 7→ hi

define morphisms of complexes φ· and ψ · between the complex 0

· · · → H i → H i+1 → · · · and the complex di

· · · → H i ⊕ B i ⊕ B i+1 → H i+1 ⊕ B i+1 ⊕ B i+2 → · · · . We have ψ · φ· = id. Consider the homomorphisms Gi : H i ⊕ B i ⊕ B i+1 → H i−1 ⊕ B i−1 ⊕ B i ,

(hi , bi , bi+1 ) 7→ (0, 0, bi ).

We have di−1 Gi + Gi+1 di = id − φi ψ i . Our assertion follows.

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Proposition 6.4.12. Let A be a ring, and let K · and L· be two bounded above complexes of A-modules. Then we have a biregular spectral sequence M i · j · p+q E2pq = TorA (K · ⊗L L· ). −p (H (K ), H (L )) ⇒ H i+j=q

Proof. each i,

Choose a projective Cartan–Eilenberg resolution P ·· of K · . For

·,i ·,i ·,i−1 ·,i ·,i−1 ·,i (P ·,i , d·,i , d1 ), (ker d·,i , d1 ) 1 ), (ker d2 , d1 ), (im d2 2 /im d2

are projective resolutions of i−1 i K i , ker diK · , im di−1 K · , ker dK · /im dK · ,

respectively. Moreover, the short exact sequences ·,i 0 → ker d·,i → im d·,i 2 →P 2 → 0,

·,i ·,i−1 0 → im d·,i−1 → ker d·,i →0 2 2 → ker d2 /im d2

are split. Choose a projective Cartan–Eilenberg resolution Q·· for L· . Consider the bicomplex M C pq = P ki ⊗A Qlj . k+l=p, i+j=q

We have a biregular spectral sequence

q E2pq = HIp HII (C ·· ) ⇒ H p+q (K · ⊗L L· ).

By Lemma 6.4.11, for each fixed k, the complex

· · · → P ki → P k,i+1 → · · ·

is homotopically equivalent to the complex 0

i+1 i · · · → HII (P k· ) → HII (P k· ) → · · · ,

and for each fixed l, the complex

· · · → Qlj → Ql,j+1 → · · ·

is homotopically equivalent to the complex 0

j j+1 · · · → HII (Ql· ) → HII (Ql· ) → · · · .

So we have

H q (P k· ⊗A Ql· ) ∼ = i HII (P ·· )

j HII (Q·· )

M

i+j=q

j i HII (P k· ) ⊗A HII (Ql· ).

But and are projective resolutions of H i (K · ) and H j (L· ), respectively. It follows that M H p H q (C ·· ) ∼ TorA (H i (K · ), H j (L· )). = I

−p

II

i+j=q

Our assertion follows.

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6.5

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Way-out Functors

([Hartshorne (1966)] I 7.) Let A and B be abelian categories, A 0 a thick abelian subcategory of A , D∗ (A ) one of the following derived categories + − b D(A ), D+ (A ), D− (A ), Db (A ), DA 0 (A ), DA 0 (A ), DA 0 (A ), DA 0 (A ),

and F : D∗ (A ) → D(B) a covariant (resp. contravariant) exact functor. We say F is way-out right if for any integer N , there exists an integer M such that for any X · ∈ ob D∗ (A ) with H i (X · ) = 0 for all i < M (resp. i > M ), we have H i (F (X · )) = 0 for all i < N . We say F is way-out left if for any integer N , there exists an integer M such that for any X · ∈ ob D∗ (A ) with H i (X · ) = 0 for all i > M (resp. i < M ), we have H i (F (X · )) = 0 for all i > N . Proposition 6.5.1. Let F : A → B be a left exact functor. Suppose that A has enough injective objects. Then RF : D+ (A ) → D(B) is way-out right. If RF has finite cohomological dimension, then RF is also way-out left. Proof. For any integer N , let X · be an object in D+ (A ) such that H i (X · ) = 0 for all i < N , then we can find a complex I · of injective objects in A with I i = 0 for all i < N such that I · is isomorphic to X · in D+ (A ). We have Ri F (X · ) ∼ = H i (F (I · )) = 0 for all i < N . So RF is way-out right. Suppose that RF has finite cohomological dimension, and let n be a nonnegative integer such that Ri F (X) = 0 for any i > n and any X ∈ ob A . For any object X · in D(A ), let X · → X 0· be a quasi-isomorphism such that X 0· is a complex of RF -acyclic objects in A . Given an integer N , if H i (X · ) = 0 for all i > N − n, then 0 → X 0N −n → X 0N −n+1 → · · · N −n is an RF -acyclic resolution of ker dX 0· . So we have −n ∼ i 0· ∼ i+N −n (F (X 0· )) ∼ Ri F (ker dN = Ri+N −n F (X · ) X 0· ) = H (F (X )[N − n]) = H −n i · for all i > 0. Since Ri F (ker dN X 0· ) = 0 for all i > n, we have R F (X ) = 0 for all i > N . Hence RF is way-out left.

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Given a complex X · of objects in A , consider the following truncated complexes: τ≤n X · = (· · · → X n−2 → X n−1 → ker dnX · → 0),

n+1 τ≥n X · = (0 → X n /im dn−1 → X n+2 → · · · ), X· → X 0 τ≤n X · = (· · · → X n−1 → X n → im dn+1 X · → 0),

0 n n+1 τ≥n X · = (0 → im dn−1 → · · · ). X· → X → X

We have quasi-isomorphisms

0 τ≤n X · → τ≤n X ·,

0 τ≥n X · → τ≥n X · ,

and the following short exact sequences of complexes 0 X · → 0, 0 → τ≤n X · → X · → τ≥n+1

0 0 → H n (X · )[−n] → τ≥n X · → τ≥n+1 X · → 0,

0 X · → τ≤n X · → H n (X · )[−n] → 0. 0 → τ≤n−1

Note that τ≤n and τ≥n transform quasi-isomorphisms to quasiisomorphisms. So they can be defined on D(A ). We have the following distinguished triangles in D(A ): τ≤n X · → X · → τ≥n+1 X · →,

H n (X · )[−n] → τ≥n X · → τ≥n+1 X · →,

τ≤n−1 X · → τ≤n X · → H n (X · )[−n] → .

Sometimes we also consider the following truncated complexes: σ≤n X · = (· · · → X n−2 → X n−1 → X n → 0),

σ≥n X · = (0 → X n → X n+1 → X n+2 → · · · ).

But σ≤n and σ≥n do not transform quasi-isomorphisms to quasiisomorphisms, and are not defined on D(A ). Proposition 6.5.2. Let A and B be abelian categories, A 0 a thick abelian + subcategory of A , F and G exact functors from DA 0 (A ), or DA 0 (A ), or − DA 0 (A ) to D(B), and ξ : F → G a morphism of functors. (i) If ξ(X) : F (X) → G(X) is an isomorphism for any X ∈ ob A 0 , then b ξ(X · ) : F (X · ) → G(X · ) is an isomorphism for any X · ∈ ob DA 0 (A ). (ii) If ξ(X) : F (X) → G(X) is an isomorphism for any X ∈ ob A 0 , and F and G are way-out right (resp. left), then ξ(X · ) : F (X · ) → G(X · ) is an + − isomorphism for any X · ∈ ob DA 0 (A ) (resp. ob DA 0 (A )). (iii) If ξ(X) : F (X) → G(X) is an isomorphism for any X ∈ ob A 0 , and F and G are way-out in both directions, then ξ(X · ) : F (X · ) → G(X · ) is an isomorphism for any X · ∈ ob DA 0 (A ).

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Proof. b · (i) For any X · ∈ ob DA 0 (A ), we have τ≥n X = 0 for sufficiently large · n. So ξ(τ≥n X ) is an isomorphism for large n. ξ induces a morphism of distinguished triangles F (H n−1 (X · )[−(n − 1)]) → F (τ≥n−1 X · ) → F (τ≥n X · ) → ξ ↓ ξ ↓ ξ ↓ G(H n−1 (X · )[−(n − 1)]) → G(τ≥n−1 X · ) → G(τ≥n X · ) → .

Since H n−1 (X · ) ∈ ob A 0 , ξ(H n−1 (X · )[−(n − 1)]) is an isomorphism by our assumption. Suppose we have shown that ξ(τ≥n X · ) is an isomorphism. Then by 6.1.1 (iii) and (TR 2), ξ(τ≥n−1 X · ) is also an isomorphism. It follows that ξ(τ≥n X · ) are isomorphisms for all n. We have τ≥n X · ∼ = X · for · n 0. So ξ(X ) is an isomorphism. + (ii) Suppose that F and G are way-out right. Given X · ∈ ob DA 0 (A ) and an integer j, let us prove that H j (ξ(X · )) : H j (F (X)) → H j (G(X))

is an isomorphism. There exists an integer M such that for any Y · ∈ + i · i · ob DA 0 (A ) with H (Y ) = 0 for all i < M , we have H (F (Y )) = H i (G(Y · )) = 0 for all i < j + 1. In particular, we have H i (F (τ≥M X · )) = H i (G(τ≥M X · )) = 0 for i = j − 1, j. We have a distinguished triangle

F (τ≤M−1 X · ) → F (X · ) → F (τ≥M X · ) → .

From the long exact sequence of cohomology objects associated to this distinguished triangle, we get H j (F (τ≤M−1 X · )) ∼ = H j (F (X · )). Similarly, we have H j (G(τ≤M−1 X · )) ∼ = H j (G(X · )). By (i), ξ induces an isomorphism H j (F (τ≤M−1 X · )) ∼ = H j (G(τ≤M−1 X · )). j · j · ∼ So ξ induces an isomorphism H (F (X )) = H (G(X )).

(iii) For any X · ∈ ob DA 0 (A ), ξ induces a morphism of distinguished triangles F (τ≤0 X · ) → F (X · ) → F (τ≥1 X · ) → ξ ↓ ξ ↓ ξ ↓ G(τ≤0 X · ) → G(X · ) → G(τ≥1 X · ) →

By (ii), ξ(τ≤0 X · ) and ξ(τ≥1 X · ) are isomorphisms. It follows that ξ(X · ) is an isomorphism.

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Proposition 6.5.3. Let A and B be abelian categories, A 0 and B 0 thick abelian subcategories of A and B, respectively, F an exact functor from + − DA 0 (A ), or DA 0 (A ), or DA 0 (A ) to D(B). (i) If F (X) ∈ ob DB0 (B) for any X ∈ ob A 0 , then F (X · ) ∈ ob DB0 (B) b for any X · ∈ ob DA 0 (A ). (ii) If F (X) ∈ ob DB0 (B) for any X ∈ ob A 0 and F is way-out right + (resp. left), then F (X · ) ∈ ob DB0 (B) for any X · ∈ ob DA 0 (A ) (resp. − · X ∈ ob DA 0 (A )). (iii) If F (X) ∈ ob DB0 (B) for any X ∈ ob A 0 and F is way-out in both directions, then F (X · ) ∈ ob DB0 (B) for any X · ∈ ob DA 0 (A ). Proposition 6.5.4. Let X be a scheme, A a ring, and F · a bounded complex of sheaves of A-modules on X. The following conditions are equivalent: (i) There exists a quasi-isomorphism F 0· → F · such that F 0· is a bounded complex of flat sheaves of A-modules. − − (ii) The functor F · ⊗L A − : D (X, A) → D (X, A) is way-out in both directions. (iii) There exists an integer n such that Tori (F · , G ) = 0 for any i > n and any sheaf of A-modules G on X. (iv) F · has finite Tor-dimension. Let f : X → Y be a morphism of schemes, F a sheaf of A-modules on X, and G a sheaf of A-modules on Y . We have a canonical morphism G ⊗A f∗ F → f∗ (f ∗ G ⊗ F ). Suppose one of the following conditions hold: (a) Rf∗ has finite cohomological dimension, F · ∈ ob D− (X, A) and G · ∈ ob D− (Y, A). (b) G · ∈ ob Db (Y, A) has finite Tor-dimension, and F · ∈ ob D+ (X, A). (c) Rf∗ has finite cohomological dimension, F · ∈ ob D(X, A), and G · ∈ ob Db (Y, A) has finite Tor-dimension. Then we have a canonical morphism · ∗ · L · G · ⊗L A Rf∗ F → Rf∗ (f G ⊗A F ).

For example, under the condition (a), let F · → I · (resp. P · → G · , resp. f ∗ P · ⊗A I · → J · ) be a quasi-isomorphism such that I · (resp. P · , resp. J · ) is a bounded above complex of Rf∗ -acyclic sheaves (resp. flat sheaves, resp. Rf∗ -acyclic sheaves) of A-modules on X (resp. on Y , resp. on X).

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The above canonical morphism is the composite · ∼ · · G · ⊗L A Rf∗ F = P ⊗A f∗ I

→ f∗ (f ∗ P ⊗A I · )

→ f∗ J · · ∼ = Rf∗ (f ∗ G ⊗L A F ). Proposition 6.5.5. Let A be a ring and let f : X → Y be a morphism of schemes. Suppose Rf∗ has finite cohomological dimension. For any F · ∈ ob D− (X, A) and any G · ∈ ob D− (Y, A) such that H i (G · ) are locally constant sheaves, we have a canonical isomorphism ∼ =

· · ∗ · L G · ⊗L A Rf∗ F → Rf∗ (f G ⊗A F ). · ∗ L · Proof. One can verify − ⊗L A Rf∗ F and Rf∗ (f − ⊗A F ) are way-out left functors. So it suffices to prove · ∼ ∗ L · G ⊗L A Rf∗ F = Rf∗ (f G ⊗A F )

for any locally constant sheaf G of A-modules. The problem is local with respect to the etale topology on Y . We may assume that G is a constant sheaf, say associated to an A-module M . Let · · · → L−1 → L0 → 0 be a resolution of M by free A-modules, and let F · → F 0· be a quasiisomorphism such that F 0· is a bounded above complex of Rf∗ -acyclic sheaves. We have · ∼ · 0· G ⊗L A Rf∗ F = L ⊗A f∗ F , f ∗ G ⊗L F · ∼ = f ∗ L· ⊗A F 0· . A

Since each Li is free and each F 0j is Rf∗ -acyclic, each f ∗ Li ⊗A F 0j is Rf∗ -acyclic. So we have · ∼ ∗ · 0· Rf∗ (f ∗ G ⊗L A F ) = f∗ (f L ⊗A F ).

Since Li are free, the canonical morphism L· ⊗A f∗ F 0· → f∗ (f ∗ L· ⊗A F 0· ) is an isomorphism. Hence · ∼ ∗ · L · L· ⊗L A Rf∗ F = Rf∗ (f L ⊗A F ).

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Corollary 6.5.6. Let A be a ring and let f : X → Y be a morphism of schemes such that Rf∗ has finite cohomological dimension. Then for any complex F · of sheaves of A-modules on X with finite Tor-dimension, Rf∗ F · also has finite Tor-dimension. Moreover, for any G · ∈ ob D(Y, A) such that H i (G · ) are locally constant sheaves, we have · ∼ ∗ · L · G · ⊗L A Rf∗ F = Rf∗ (f G ⊗A F ).

Proof. Let G be a constant sheaf of A-modules on Y . Since F · has finite Tor-dimension, there exists an integer n such that Tori (f ∗ G , F · ) = · 0 for all i > n, that is, H j (f ∗ G ⊗L A F ) = 0 for all j < −n. Thus j ∗ L · R f∗ (f G ⊗A F ) = 0 for all j < −n. By 6.5.5, we have · ∼ j L · ∼ · Rj f∗ (f ∗ G ⊗L A F ) = H (G ⊗ Rf∗ F ) = Tor−j (G , Rf∗ F ).

So Tori (G , Rf∗ F · ) = 0 for all i > n. Hence Rf∗ F · has finite Tor· ∗ L · dimension. The functors − ⊗L A Rf∗ F and Rf∗ (f − ⊗A F ) are way-out in both directions. By 6.5.2 (ii) and 6.5.5, we have · · ∼ ∗ · L G · ⊗L A Rf∗ F = Rf∗ (f G ⊗A F )

for any G · ∈ ob D(Y, A) such that H i (G · ) are locally constant sheaves. Corollary 6.5.7. Let f : X → Y be a morphism such that Rf∗ has finite cohomological dimension, and let A → B be a homomorphism of rings. For any F · ∈ ob D− (X, A) and any object M · in the derived category D− (B) of the category of B-modules, we have a canonical isomorphism ∼ =

· · L · M · ⊗L A Rf∗ F → Rf∗ (M ⊗A F )

in D− (Y, B). Proof. (We cannot apply 6.5.5 directly since it gives an isomorphism in D− (Y, A), and not in D− (Y, B).) Let F 0· be a bounded above complex of flat sheaves of A-modules on X quasi-isomorphic to F · and let C · (F 0i ) be the Godement resolution of F 0i . Choose an integer d such that Ri f∗ F = 0 for all i > d and all sheaves F on X. Then τ≤d C · (F 0i ) is a resolution of F 0i by Rf∗ -acyclic sheaves. The complex associated to the bicomplex τ≤d C · (F 0· ) is a bounded above complex of Rf∗ -acyclic sheaves quasi-isomorphic to F · . Denote this complex by F 00· . We define a morphism · · L · M · ⊗L A Rf∗ F → Rf∗ (M ⊗A F )

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in D− (Y, B) to be the composite · ∼ · L 00· M · ⊗L A Rf∗ F = M ⊗A f∗ F

→ M · ⊗A f∗ F 00·

→ f∗ (f ∗ M · ⊗A F 00· )

→ Rf∗ (f ∗ M · ⊗A F 00· ) · ∼ = Rf∗ (M · ⊗L A F ). To prove that it is an isomorphism, it suffices to show that it coincides with the morphism defined in 6.5.5. This follows from the commutativity of the following diagram, where L· is a bounded above complex of free A-modules such that we have a quasi-isomorphism L· → M · :

∼ = · ∼ · M · ⊗L = L· ⊗A f∗ F 00· → f∗ (f ∗ L· ⊗A F 00· ) ∼ = Rf∗ (f ∗ L· ⊗A F 00· ) ∼ = Rf∗ (M · ⊗L A Rf∗ F AF )

k

↓

↓

↓

k

· · 00· · M · ⊗L → f∗ (f ∗ M · ⊗A F 00· ) → Rf∗ (f ∗ M · ⊗A F 00· ) ∼ = Rf∗ (M · ⊗L A Rf∗ F → M ⊗A f∗ F A F ).

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Base Change Theorems

7.1

Divisors

Let X be a noetherian scheme. We define the dimension dim X of X to be the supremum of all integers n so that there exists a chain Z0 ⊂ Z1 ⊂ · · · ⊂ Zn of distinct irreducible closed subsets in X. If X = Spec A for a noetherian ring A, then dim X coincides with the Krull dimension dim A of A. If Z is an irreducible closed subset of X, we define the codimension codim (Z, X) of Z in X to be the supremum of all integers n so that there exists a chain Z0 ⊂ Z1 ⊂ · · · ⊂ Zn of distinct irreducible closed subsets in X with Z0 = Z. For any closed subset Y in X, we define the codimension codim(Y, X) of Y in X to be codim(Y, X) = inf Z⊂Y codim(Z, X), where Z goes over the set of irreducible closed subset of Y . If X = Spec A and Y = V (p) for some prime ideal p of A, then codim(Y, X) coincides with the height ht p of p. A noetherian scheme X is called regular in codimension one if for any x ∈ X with dim OX,x = 1, OX,x is regular. Recall that a local noetherian integral domain of dimension 1 is regular if and only if it is normal ([Atiyah and Macdonald (1969)] 9.2). So any normal noetherian scheme is regular in codimension 1. Suppose that X is an integral noetherian scheme regular in codimension one. A prime divisor on X is an integral closed subscheme of X of codimension 1. A Weil divisor on X is an element in the free abelian group P Div(X) generated by prime divisors. We write a divisor as D = i ni Yi , 311

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where Yi are prime divisors, ni are integers, and ni are nonzero only for finitely many i. If ni ≥ 0 for all i, we say that D is an effective divisor. Let Y be a prime divisor, and let η be its generic point. Then dim OX,η = 1. Since X is regular in codimension 1, OX,η is a discrete valuation ring. It defines a discrete valuation vY on the function field K of X. Lemma 7.1.1. Let X be an integral noetherian scheme regular in codimension one, and let K be the function field of X. For any f ∈ K ∗ , there are only finitely many prime divisors Y such that vY (f ) 6= 0. Proof. Let U = Spec A be a nonempty affine open subscheme of X so that f ∈ OX (U ) = A, let Y be a prime divisor of X, and let η be the generic point of Y . If η 6∈ U , then Y ⊂ X − U . Since Y has codimension 1, Y must be an irreducible component of X − U . Since X − U has only finitely many irreducible components, there are only finitely many prime divisors Y with η 6∈ U . Suppose η ∈ U and vY (f ) 6= 0. Let p be the prime ideal of A corresponding to η. Since vY (f ) 6= 0 and f ∈ A, we have f ∈ p. It follows that Y ∩ U = V (p) ⊂ V (f ). As Y ∩ U has codimension 1 in U , Y ∩ U must be an irreducible component of V (f ). Since V (f ) has only finitely many irreducible components, there are only finitely many prime divisors Y such that vY (f ) 6= 0. Let X be an integral noetherian scheme regular in codimension one and let K be its function field. For any f ∈ K ∗ , we define the divisor (f ) of f to be X (f ) = vY (f )Y, Y

where Y goes over the set of prime divisors of X. For any f, g ∈ K ∗ , we have (f g) = (f ) + (g). A divisor on X is called principal if it is of the form (f ) for some f ∈ K ∗ . Principal divisors form a subgroup of the group of divisors Div(X). The quotient group Cl(X) is called the divisor class group of X. Proposition 7.1.2. Let A be a normal noetherian integral domain, and let X = Spec A. Then Cl(X) = 0 if and only if A is a unique factorization domain.

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Proof. Suppose A is a unique factorization domain. Let Y be a prime divisor of X, and let p be the prime ideal of A corresponding to the generic point of Y . Then ht p = 1. By [Matsumura (1970)] (19.A) Theorem 47, we have p = f A for some f ∈ A. For any prime ideal q of A with ht q = 1, denote by vq the valuation defined by q. If vq (f ) 6= 0, then we have f ∈ q, and hence p ⊂ q. We then must have p = q. It follows that the principal divisor (f ) coincides with the prime divisor Y . So every prime divisor of X is principal and Cl(X) = 0. Conversely, suppose Cl(X) = 0. For any prime ideal p with ht p = 1, let Y be the prime divisor with generic point p. Then Y is a principal divisor, say Y = (f ) for some nonzero element f in the fraction field K of A. For any prime ideal q with ht q = 1, we have 1 if q = p, vq (f ) = 0 if q 6= p. In particular, we have f∈

\

Aq ,

ht q=1

and hence f ∈ A by [Matsumura (1970)] (17.H) Theorem 38. As vp (f ) = 1, we have f ∈ p. For any g ∈ p, we have vp (g) ≥ 1 and vq (g) ≥ 0 for any q with ht q = 1. It follows that vq ( fg ) ≥ 0 for any q with ht q = 1. So we have \ g ∈ Aq = A. f ht q=1

Hence g ∈ f A. We thus have p = f A. So every prime ideal of height 1 is principal. By [Matsumura (1970)] (19.A) Theorem 47, A is a unique factorization domain. Let X be an integral scheme, and let K be its function field. Denote by KX the constant Zariski sheaf on X associated to K, and by KX∗ the constant Zariski sheaf associated to K ∗ . Any nonempty open subset U of X is irreducible and hence connected. So we have KX∗ (U ) = K ∗ . In particular, KX∗ (X) → KX∗ (U ) is surjective. Hence KX∗ is a flasque sheaf (with respect to the Zariski topology) and we have H q (X, KX∗ ) = 0 for all q ≥ 1. We have a monomorphism ∗ → KX∗ . OX

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∗ A Cartier divisor on X is an element in Γ(X, KX∗ /OX ). A Cartier divisor can be represented as (fi , Ui ), where {Ui } is an open covering of X, fi ∈ ∗ Γ(Ui , KX∗ ), and ffji ∈ Γ(Ui ∩ Uj , OX ) for all i and j. If fi can be chosen to lie in Γ(Ui , OX ), then we say that the Cartier divisor is effective. A Cartier divisor is called principal if it lies in the image of the canonical homomorphism ∗ Γ(X, KX∗ ) → Γ(X, KX∗ /OX ).

The cokernel of this homomorphism is called the Cartier divisor class group, and is denoted by CaCl(X). Proposition 7.1.3. Let X be an integral noetherian scheme. Suppose OX,x are unique factorization domains for all x ∈ X. Then we have an isomorphism ∼ =

∗ Γ(X, KX∗ /OX ) → Div(X),

and it induces an isomorphism between the group of principal Cartier divisors and the group of principal Weil divisors. So it induces an isomorphism ∼ =

CaCl(X) → Cl(X)

between the Cartier divisor class group and the Weil divisor class group. Proof. Let D = {(fi , Ui )} be a Cartier divisor, where {Ui } is an open ∗ covering of X, fi ∈ Γ(Ui , KX∗ ), and ffji ∈ Γ(Ui ∩ Uj , OX ) for all i and j. For any prime divisor Y , let Ui be an open subset such that Ui ∩ Y 6= Ø. We define vY (D) = vY (fi ). Note that vY (D) is independent of the choice of the open subset Ui with nonempty intersection with Y . Define the Weil divisor associated to D to P be Y vY (D)Y . We thus get a homomorphism ∗ Γ(X, KX∗ /OX ) → Div(X).

First let us prove that it is injective. Suppose D = {(fi , Ui )} is mapped to the trivial Weil divisor. We need to show that D is the trivial Cartier divisor. We may assume that each Ui = Spec Ai is affine. Then for any prime ideal p of Ai with ht p = 1, we have vp (fi ) = 0. By [Matsumura ∗ (1970)] (17.H) Theorem 38, we have fi ∈ OX (Ui ). Hence D is trivial. Next ∗ we prove that the homomorphism Γ(X, KX∗ /OX ) → Div(X) is surjective under the assumption that OX,x is a unique factorization domain for any P x ∈ X. For any Weil divisor D = Y nY Y on X, let X Dx = nY (Y ⊗OX OX,x ) Y

etale

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be the divisor on Spec OX,x induced by D. By 7.1.2, we have Dx = (fx ) for some fx ∈ OX,x . Let Ux be an open neighborhood of x so that fx can be extended to a section in Γ(Ux , OX ), which we still denote by fx . There are only finitely many prime divisors Y with nonempty intersection with Ux such that both vY (fx ) and nY are nonzero. Shrinking Ux , we may assume X nY (Ux ∩ Y ) = (fx ). Y

For any prime divisor Y with nonempty intersection with Ux ∩ Ux0 , we have f x vY = 0. fx0 So we have fx ∗ ∈ OX (Ux ∩ Ux0 ) fx0 by [Matsumura (1970)] (17.H) Theorem 38. Thus {(fx , Ux )} is a Cartier divisor. This Cartier divisor is mapped to D under the homomorphism ∗ Γ(X, KX∗ /OX ) → Div(X). Hence the homomorphism is surjective. Remark 7.1.4. Let X be an integral noetherian scheme. Suppose OX,x are regular for all x ∈ X, that is, X is a regular scheme. Then OX,x are unique factorization domains for all x by [Matsumura (1970)] (19.B) Theorem 48. By 7.1.3, we have an isomorphism CaCl(X) ∼ = Cl(X). Proposition 7.1.5. Let X be an integral scheme. We have a canonical isomorphism ∼ =

CaCl(X) → Pic(X). Proof.

We have an exact sequence ∗ ∗ → 0. 0 → OX → KX∗ → KX∗ /OX

Since H 1 (X, KX∗ ) = 0, we have an exact sequence

∗ ∗ Γ(X, KX∗ ) → Γ(X, KX∗ /OX ) → H 1 (X, OX ) → 0.

So we have an isomorphism ∗ CaCl(X) ∼ ). = H 1 (X, OX

But ∗ ∼ H 1 (X, OX ) = Pic(X).

So we have CaCl(X) ∼ = Pic(X).

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A more direct proof is as follows. Let D = {(fi , Ui )} be a Cartier divisor, where {Ui } is an open covering of X, fi ∈ Γ(Ui , KX∗ ), and ffji ∈ ∗ Γ(Ui ∩ Uj , OX ) for all i and j. We have fi−1 OUi ∩Uj = fj−1 OUi ∩Uj .

So we have an OX -submodule L (D) of K so that L (D)|Ui = fi−1 OUi . L (D) is an invertible OX -module, and L (D1 + D2 ) ∼ = L (D1 ) ⊗OX L (D2 ) for all Cartier divisors D1 and D2 . So D → L (D) defines a homomorphism ∗ Γ(X, KX∗ /OX ) → Pic(X).

One can check that L (D) is isomorphic to OX if and only if D is a principal Cartier divisor. We thus have a monomorphism CaCl(X) ,→ Pic(X). For any invertible OX -module L , we can find an open covering {Ui } of X and ei ∈ Γ(Ui , L ) so that we have isomorphisms ∼ =

OX |Ui → L |Ui ,

s 7→ sei .

We have ei |Ui ∩Uj = fij ej |Ui ∩Uj ∗ for some fij ∈ Γ(Ui ∩ Uj , OX ). Fix a nonempty open subset Ui0 in the open covering. Let D be the Cartier divisor D = {(fi , Ui )} defined by

fi = fi0 i . Here we regard fi0 i as a section of KX∗ over Ui . Then L ∼ = L (D). This ∗ proves that the homomorphism Γ(X, KX∗ /OX ) → Pic(X) is surjective. Let X be a smooth algebraic curve over a field k. Then a prime Weil divisor on X is a Zariski closed point of X with the reduced closed subscheme structure. Let |X| be the set of all Zariski closed points of X. For any x ∈ X, let k(x) be the residue field of OX,x . We define the degree of x to be deg(x) = [k(x) : k].

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For any Weil divisor D =

P

x∈|X| nx x,

deg(D) =

X

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317

we define the degree of D to be nx deg(x).

x∈|X|

At the end of 8.1, we give a proof of the following fact. Proposition 7.1.6. Let X be a smooth proper algebraic curve over a field k, and let K(X) be the function field of X. For any nonzero f ∈ K(X), we have deg(f ) = 0, where (f ) is the principal divisor defined by f . Suppose that X is a smooth proper algebraic curve over a field k. Taking degree defines a homomorphism deg : Div(X) → Z, and it induces a homomorphism deg : Cl(X) → Z. For any invertible OX -module L , choose a Weil divisor D on X such that L ∼ = L (D), where L (D) is the invertible OX -module defined in the proof of 7.1.5 using the Cartier divisor corresponding to the Weil divisor D. We define the degree of L to be deg(L ) = deg(D). Then taking degree defines a homomorphism deg : Pic(X) → Z. 7.2

Cohomology of Curves

([SGA 4] IX 4.) Proposition 7.2.1 (Kummer’s theory). Let X be a scheme and let n be a positive integer invertible on X. Then the morphism of etale sheaves ∗ ∗ n : OX → OX , et et

s 7→ sn

is surjective. Let µn,X be its kernel. We thus have a short exact sequence n

∗ ∗ 0 → µn,X → OX → OX → 0. et et

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Proof. Let U = Spec A be an affine etale X-scheme and let a ∈ ∗ OX (U ) = A∗ be a section. The canonical morphism et V = Spec A[t]/(tn − a) → U = Spec A is etale by 2.3.3 since tn − a and its derivative ntn−1 generate A[t]. One can show that the fibers of V → U are nonempty and hence V → U is surjective. So {V → U } is an etale covering of U . The restriction of the ∗ ∗ section a to V has an n-th root t. So n : OX → OX is surjective. et et Proposition 7.2.2. Let X be a scheme and let n be a positive integer invertible on X. The sheaf µn,X is locally constant. When X is a scheme over a strictly local ring A such that n is invertible in A, µn,X is isomorphic to the constant sheaf Z/n. Proof. Note that for any etale X-scheme U , we have a one-to-one correspondence {s ∈ OU (U )∗ |sn = 1} ∼ = HomX (U, Spec OX [t]/(tn − 1)). So µn,X is represented by the X-scheme Spec OX [t]/(tn − 1). This scheme is finite and etale over X. So µn,X is locally constant by 5.8.1 (i). If A is strictly henselian and n is invertible in A, then tn − 1 splits into a product of linear polynomials in A[t] by 2.8.3 (v). For any A-scheme X, Spec OX [t]/(tn − 1) is then a trivial etale covering of X of degree n. Hence µn,X is isomorphic to the constant sheaf Z/n. Proposition 7.2.3 (Artin–Schreier’s theory). Let p be a prime number and let X be a scheme such that p · 1 = 0 in Γ(X, OX ). Then the morphism of etale sheaves s 7→ sp − s

℘ : OXet → OXet ,

is surjective. Its kernel is isomorphic to Z/p. We thus have a short exact sequence ℘

0 → Z/p → OXet → OXet → 0. Proof. Let U = Spec A be an affine etale X-scheme and let a ∈ OXet (U ) = A be a section. The canonical morphism V = Spec A[t]/(tp − t − a) → U = Spec A is etale by 2.3.3 since the derivative −1 of tp − t − a generate A[t]. One can show the fibers of V → U are nonempty and hence V → U is surjective.

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So {V → U } is an etale covering of U . The restriction of the section a to V is the image of t under the homomorphism ℘ : OXet (V ) = A[t]/(tp − t − a) → OXet (V ) = A[t]/(tp − t − a). So ℘ : OXet → OXet is surjective. Note that for any etale X-scheme U , we have a one-to-one correspondence ∼ HomX (U, Spec OX [t]/(tp − t)), {s ∈ OU (U )|sp − s = 0} =

that is, ker ℘ is represented by the X-scheme Spec OX [t]/(tp − t). We have Y tp − t = (t − i) i∈Z/p

on X. It follows that ker ℘ ∼ = Z/p.

Lemma 7.2.4. Let X be a noetherian scheme and let F be a sheaf on X. The following conditions are equivalent: (i) For every non-closed point y in X, we have Fy¯ = 0. (ii) The canonical morphism Y F → ix∗ i∗x F x∈|X|

induces an isomorphism

F ∼ =

M

ix∗ i∗x F ,

x∈|X|

where |X| is the set of Zariski closed points in X, and ix : Spec k(x) → X are the closed immersions. When F satisfies these conditions, we say that F is a skyscraper sheaf. Proof. Suppose (i) holds. Let f : U → X be an etale morphism of finite type. For any s ∈ F (U ), if u ∈ U is a point in the support of s, then f (u) must be a closed point of X. Since f is quasi-finite, u is closed in f −1 (f (u)). Hence u is a closed point of U . Since U is noetherian, the support of s consists of finitely many closed points of U . It follows that the image of the canonical morphism Y ix∗ i∗x F F → is contained in the morphism

L

x∈|X|

∗ x∈|X| ix∗ ix F .

F →

One shows that under the condition (i), M

x∈|X|

ix∗ i∗x F

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induces an isomorphism on stalks at every point of X, and hence it is an isomorphism. Here we use the formula Fx¯ if y = x, ∗ ∼ (ix∗ ix F )y¯ = 0 if y 6= x for any x ∈ |X| and any y ∈ X. This proves (i)⇒(ii). It is clear that (ii)⇒(i). Lemma 7.2.5. Let X be a noetherian scheme with dim X ≤ 1, let η1 , . . . , ηm be all the points in X satisfying dim {ηi } = 1 (i = 1, . . . , m), Q let R = m i=1 OX,ηi , and let j : Spec R → X be the canonical morphism. (i) For any sheaf G on Spec R and any q ≥ 1, Rq j∗ G is a skyscraper sheaf. (ii) For any sheaf F on X, the kernel and cokernel of the canonical morphism F → j∗ j ∗ F are skyscraper sheaves. Proof. (i) For each i, let O˜X,¯ηi be the strict henselization of OX,ηi . By 5.9.5, we have (Rq j∗ G )η¯i ∼ = H q (Spec (R ⊗OX O˜X,¯ηi ), G ). One can show R ⊗OX O˜X,¯ηi ∼ = O˜X,¯ηi . By 5.7.3, we have H q (Spec O˜X,¯ηi , G ) = 0 for any q ≥ 1. So (Rq j∗ G )η¯i = 0 for any q ≥ 1. But ηi (i = 1, . . . , m) are all the non-closed points of X. So Rq j∗ G is a skyscraper sheaf for any q ≥ 1. (ii) We have (j∗ j ∗ F )η¯i ∼ = H 0 (Spec O˜X,¯ηi , j ∗ F ) ∼ = Fη¯i . = H 0 (Spec (R ⊗OX O˜X,¯ηi ), j ∗ F ) ∼ So the canonical morphism F → j∗ j ∗ F induces isomorphisms on stalks at η¯i . It follows that its kernel and cokernel are skyscraper sheaves. Lemma 7.2.6. Let X be a scheme of finite type over a separably closed field k. Then for any skyscraper sheaf F on X, we have H q (X, F ) = 0 for any q ≥ 1.

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Proof.

We have H q (X, F ) ∼ = H q (X, ∼ =

M

M

etale

321

ix∗ i∗x F )

x∈|X|

H q (X, ix∗ i∗x F )

x∈|X|

∼ =

M

H q (Spec k(x), i∗x F ).

x∈|X|

For any closed point x, the residue field k(x) is separably closed. H q (Spec k(x), i∗x F ) = 0 for any q ≥ 1. Our assertion follows.

So

Theorem 7.2.7. Let X be a reduced scheme of finite type over a separably closed field k of characteristic p with dim X ≤ 1. (i) For any torsion sheaf F on X, we have H q (X, F ) = 0 for any q ≥ 3. ∗ ∗ (ii) H 2 (X, OX ) and H 3 (X, OX ) are p-torsion groups, and et et q ∗ H (X, OXet ) = 0 for any q ≥ 4. If k is algebraically closed, then ∗ H q (X, OX ) = 0 for any q ≥ 2. et Proof. Let η1 , . . . , ηm be all the points in X satisfying dim {ηi } = 1 (i = Qm 1, . . . , m), R = i=1 OX,ηi , j : Spec R → X be the canonical morphism, F a sheaf on X, F → Rj∗ j ∗ F the canonical morphism, and F → Rj∗ j ∗ F → ∆ → a distinguished triangle. We have an exact sequence 0 → H −1 (∆) → F → j∗ j ∗ F → H 0 (∆) → 0 and Rq j∗ j ∗ F ∼ = H q (∆) for any q ≥ 1. By 7.2.5, H q (∆) are skyscraper sheaves for any q. By 7.2.6, the biregular spectral sequence E2pq = H p (X, H q (∆)) ⇒ H p+q (X, ∆) degenerates and we have H q (X, ∆) ∼ = H 0 (X, H q (∆)) for any q. Moreover, we have H q (X, Rj∗ j ∗ F ) ∼ = H q (Spec R, j ∗ F ).

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The long exact sequence of H · (X, −) associated to the above distinguished triangle is · · · → H q (X,F ) → H q (Spec R,j ∗ F ) → H 0 (X,H q (∆)) → H q+1 (X,F ) → · · · . Since X is reduced, each OX,ηi is a field of transcendental degree 1 over k. Suppose that F is a torsion sheaf. We have H q (Spec R, j ∗ F ) = 0 for any q ≥ 2 by 5.7.8 and 4.5.11 (i). So we have H q (X, F ) ∼ = H 0 (X, H q−1 (∆)) for any q ≥ 3. On the other hand, Rq j∗ j ∗ F is the sheaf associated to the presheaf V 7→ H q ((Spec R) ×X V, j ∗ F ) for any etale X-scheme V . Note that (Spec R) ×X V is a disjoint union of the spectra of some fields finite separable over some OX,ηi . Again by 5.7.8 and 4.5.11 (i), we have H q ((Spec R) ×X V, j ∗ F ) = 0 for any q ≥ 2. So Rq j∗ j ∗ F = 0 for any q ≥ 2, and hence H q (∆) = 0 for any q ≥ 2. Therefore H q (X, F ) ∼ = H 0 (X, H q−1 (∆)) = 0 for any q ≥ 3. This proves (i). ∗ ∗ and k is algebraically closed). We (resp. F = OX Suppose F = OX et et ∗ ∗ ∗ ∼ have j OXet = O(Spec R)et . By 5.7.8 and 4.5.11 (ii) (resp. 4.5.8 and 4.5.9), we have ∗ H q (Spec R, j ∗ OX )=0 et

for q = 1 and any q ≥ 3 (resp. q ≥ 1). So we have ∗ H q (X, OX )∼ = H 0 (X, H q−1 (∆)) et ∗ is the sheaf for any q ≥ 4 (resp. q ≥ 2.) On the other hand, Rq j∗ j ∗ OX et associated to the presheaf ∗ V 7→ H q ((Spec R) ×X V, O(Spec R)et )

for any etale X-scheme V . Again by 5.7.8 and 4.5.11 (ii) (resp. 4.5.8 and 4.5.9), we have ∗ H q ((Spec R) ×X V, O(Spec R)et ) = 0

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∗ for q = 1 and any q ≥ 3 (resp. q ≥ 1). So Rq j∗ j ∗ OX = 0 for q = 1 and et q any q ≥ 3 (resp. q ≥ 1), and hence H (∆) = 0 for q = 1 and any q ≥ 3 (resp. q ≥ 1). Therefore ∼ H 0 (X, H q−1 (∆)) = 0 H q (X, O ∗ ) = Xet

for any q ≥ 4 (resp. q ≥ 2). ∗ Since H 0 (X, H 1 (∆)) = 0 and H 3 (Spec R, j ∗ OX ) = 0, part of the et above long exact sequence is ∗ ∗ ∗ 0 → H 2 (X,OX ) → H 2 (Spec R,j ∗ OX ) → H 0 (X,H 2 (∆)) → H 3 (X,OX ) → 0. et et et

∗ ∗ By 4.5.11 (ii), H 2 (Spec R, j ∗ OX ) is a p-torsion group and R2 j∗ j ∗ OX et et 2 is a p-torsion sheaf. Hence H (∆) is a p-torsion sheaf. It follows that ∗ ∗ H 2 (X, OX ) and H 3 (X, OX ) are p-torsion groups. et et

Remark 7.2.8. Let X be a regular integral scheme of dimension 1, η the generic points, K = OX,η the function field of X, and j : Spec K → X the canonical morphism. Define a morphism M ∗ j∗ O(Spec → ix∗ Z K)et x∈|X|

as follows: For any connected etale X-scheme V of finite type, let η 0 be the generic point of V and let K 0 = OV,η0 . We have 0∗ ∗ (j∗ O(Spec K)et )(V ) = K , M M Z. ix∗ Z (V ) = x∈|V |

x∈|X|

Define

∗ (j∗ O(Spec K)et )(V

to be K 0∗ →

M

x∈|V |

ix∗ Z (V )

)→

M

Z,

f 7→ (vx (f )),

x∈|X|

where vx : K 0∗ → Z is the valuation at x. We can identify the sheaf L x∈|X| ix∗ Z with the sheaf of Weil divisors D defined by setting D(V ) ∗ to be the group of Weil divisors on V . The morphism j∗ O(Spec K)et → L x∈|X| ix∗ Z is simply the morphism mapping each rational function to the principle Weil divisor associated to it. This morphism is surjective and its kernel is exactly OXet . We thus have an exact sequence M ∗ ix∗ Z → 0. 0 → OXet → j∗ O(Spec K)et → x∈|X|

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Theorem 7.2.9. (i) Let X be a reduced scheme of finite type over a separably closed field k with dim X ≤ 1, and let n be a positive integer invertible in k. Then H q (X, µn,X ) = 0 for any q ≥ 3 and we have an exact sequence n

∗ ∗ 0 → H 0 (X, µn,X ) → Γ(X, OX ) → Γ(X, OX )→ n

→ H 1 (X, µn,X ) → Pic(X) → Pic(X) → H 2 (X, µn,X ) → 0.

(ii) Let X be a smooth projective irreducible curve of genus g over an algebraically closed field k, and let n be a positive integer invertible in k. Then we have  Z/n if q = 0,    2g (Z/n) if q = 1, H q (X, Z/n) ∼ =  Z/n if q = 2,   0 if q ≥ 3. Moreover, we have canonical isomorphisms  {x ∈ k|xn = 1}    {L ∈ Pic(X)|nL = 0} H q (X, µn,X ) ∼ =  Z/n   0

if if if if

q q q q

= 0, = 1, = 2, ≥ 3.

Proof. (i) By Kummer’s theory 7.2.1, we have a long exact sequence n

∗ ∗ 0 → H 0 (X, µn,X ) → H 0 (X, OX ) → H 0 (X, OX ) → H 1 (X, µn,X ) n

∗ ∗ ∗ → H 1 (X, OX ) → H 1 (X, OX ) → H 2 (X, µn,X ) → H 2 (X, Oet ) → ··· .

We have ∗ ∗ H 0 (X, OX ) = Γ(X, OX ),

∗ H 1 (X, OX ) = Pic(X).

∗ Moreover, if p = char k, then H q (X, OX ) are p-torsion groups for any q ≥ 2 q by 7.2.7, whereas H (X, µn,X ) are Z/n-modules. Since n is invertible in k, n is relatively prime to p. Our assertion follows. (ii) Since X is projective and integral, and k is algebraically closed, we ∗ have Γ(X, OX ) = k ∗ , and the homomorphism

n : k∗ → k∗ ,

x 7→ xn

is surjective. It follows from the long exact sequence in (i) that H 0 (X, µn,X ) ∼ = {x ∈ k|xn = 1}, H 1 (X, µn,X ) ∼ = Pic(X)n , 2 ∼ H (X, µn,X ) = Pic(X)/nPic(X),

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where for any abelian group A, we set An = ker(n : A → A). Let deg : Pic(X) → Z be the homomorphism mapping an invertible OX -module to its degree. It is surjective. Let Pic0 (X) be its kernel. We have a commutative diagram of short exact sequences deg

0 → Pic0 (X) → Pic(X) → Z → 0 n↓ n↓ n↓ deg

0 → Pic0 (X) → Pic(X) → Z → 0,

where all the vertical arrows are multiplication by n. By the snake lemma, we have an exact sequence 0 → Pic0 (X)n → Pic(X)n → 0

→ Pic0 (X)/nPic0 (X) → Pic(X)/nPic(X) → Z/n → 0.

Pic0 (X) is isomorphic to the group of k-points JacX (k) of the Jacobian JacX of X. Moreover the homomorphism n : JacX (k) → JacX (k) is surjective, and its kernel is isomorphic to (Z/n)2g . So we have H 1 (X, µn,X ) ∼ = Pic(X)n ∼ = Pic0 (X)n ∼ = JacX (k)n ∼ = (Z/n)2g , and H 2 (X, µn,X ) ∼ = Pic(X)/nPic(X) ∼ = Z/n. By (i), we have H q (X, µn,X ) = 0 for any q ≥ 3. This proves the second part of (ii). By 7.2.2, we have Z/n ∼ = µn,X . The first part of (ii) follows. Proposition 7.2.10. Let X be a smooth irreducible curve over a separably closed field k, and let n be a positive integer invertible in k. Then H q (X, Z/n) are finite for all q and vanish for all q ≥ 3. If X is not projective, then H q (X, Z/n) vanish for all q ≥ 2. Proof. Let k¯ be an algebraic closure of k. Then k¯ = limi ki , where ki −→ ¯ By goes over the family of finite purely inseparable extensions of k in k. 5.7.2, we have H q (X, Z/n) ∼ = H q (X ⊗k ki , Z/n), and by 5.9.3, we have ¯ Z/n) ∼ H q (X ⊗k k, H q (X ⊗k ki , Z/n). = lim −→ i

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It follows that ¯ Z/n). H q (X, Z/n) ∼ = H q (X ⊗k k, So to prove our assertion, we may assume that k is algebraically closed. We have Z/n ∼ = µn,X by 7.2.2. So we may replace Z/n by µn,X . By 7.2.9 (i), we have H q (X, µn,X ) = 0 for any q ≥ 3. Since X is irreducible and hence connected, we have H 0 (X, Z/n) ∼ = Z/n. In particular, H 0 (X, Z/n) is finite. Let X be the smooth compactification of X and let X − X = ∗ {x1 , . . . , xm }. Then Γ(X, OX ) consists of nonzero rational functions on X regular and invertible on X. Consider the homomorphism ∗ Γ(X, OX ) → Zm ,

f 7→ (vx1 (f ), . . . , vxm (f )),

where vxi (i = 1, . . . , m) denote the valuation at xi . The kernel of this homomorphism consists of rational functions on X regular everywhere, and hence is isomorphic to k ∗ . Let A be the image of this homomorphism. We have an exact sequence ∗ 0 → k ∗ → Γ(X, OX ) → A → 0.

By the snake lemma, we have an exact sequence ∗ ∗ n k ∗ /k ∗n → Γ(X, OX )/Γ(X, OX ) → A/nA → 0.

Since A is a subgroup of Zm , A/nA is finite. Since k is algebraically closed, ∗ ∗ n we have k ∗ /k ∗n = 0. It follows that Γ(X, OX )/Γ(X, OX ) is finite. The canonical restriction homomorphism Pic(X) → Pic(X),

L → L |X

is surjective. Let B be the kernel of this homomorphism. By the snake lemma, we have an exact sequence Pic(X)n → Pic(X)n → B/nB → Pic(X)/nPic(X) → Pic(X)/nPic(X) → 0. In the proof of 7.2.9, we have seen Pic(X)n ∼ = (Z/n)2g ,

Pic(X)/nPic(X) ∼ = Z/n,

where g is the genus of X. In particular, Pic(X)n and Pic(X)/nPic(X) are finite. We will show in a moment that B/nB is finite. The above exact sequence then shows that Pic(X)n and Pic(X)/nPic(X) ∼ = H 2 (X, µn,X ) are finite. To prove that B/nB is finite, consider the homomorphism Zm → B mapping (n1 , . . . , nm ) to the isomorphic class of the invertible OX -module

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associated to the Weil divisor n1 x1 + · · · + nm xm . It is surjective. Indeed, for any L ∈ B, we have L |X ∼ = OX . Let D be a Weil divisor on X so that L ∼ = L (D). Then D|X is a principle divisor, say D|X = (f )|X for some rational function f . Then D − (f ) is a Weil divisor on X supported on {x1 , . . . , xm } and L ∼ = L (D − (f )). So L lies in the image of Zm → B. Thus the homomorphism Zm → B is surjective. This implies that B/nB is finite. By 7.2.9 (i), we have an exact sequence n

n

∗ ∗ Γ(X, OX ) → Γ(X, OX ) → H 1 (X, µn,X ) → Pic(X) → Pic(X). ∗ ∗ n Combined with the finiteness of Γ(X, OX )/Γ(X, OX ) and Pic(X)n , we see 1 that H (X, µn ) is finite. If X is not projective, then X − X 6= Ø, and every Weil divisor on X is the restriction of a Weil divisor of degree 0 on X. So the restriction homomorphism

Pic0 (X) → Pic(X),

L → L |X

is surjective. We have seen in the proof of 7.2.9 that n : Pic0 (X) → Pic0 (X) is surjective. It follows that n : Pic(X) → Pic(X) is surjective. So H 2 (X, µn,X ) ∼ = Pic(X)/nPic(X) = 0.

Theorem 7.2.11. Let X be a scheme of finite type over a separably closed field k of characteristic p. If X is affine, then H q (X, Z/p) = 0 for any q ≥ 2. If X is proper, then H q (X, Z/p) = 0 for any q > dim X, and we have an exact sequence ℘

q q 0 → H q (X, Z/p) → HZar (X, OX ) → HZar (X, OX ) → 0

q q for every q, where ℘ : HZar (X, OX ) → HZar (X, OX ) is the homomorphism induced by

℘ : OX → OX ,

s 7→ sp − s.

Proof. We may assume that k is algebraically closed. By Artin–Schreier’s theory 7.2.3, we have an exact sequence ℘

0 → H 0 (X, Z/p) → H 0 (X, OXet ) → H 0 (X, OXet ) → · · · ℘

→ H q (X, Z/p) → H q (X, OXet ) → H q (X, OXet ) → · · ·

By 5.7.5, we have q H q (X, OXet ) ∼ = HZar (X, OX )

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q for all q. If X is affine, we have HZar (X, OX ) = 0 for any q ≥ 1. So q q H (X, Z/p) = 0 for any q ≥ 2. If X is proper, each HZar (X, OX ) is a finite dimensional k-vector space. The homomorphism q q ℘ : HZar (X, OX ) → HZar (X, OX )

is of the form φ − id, where

q q φ : HZar (X, OX ) → HZar (X, OX )

is the homomorphism induced by OX → OX ,

s 7→ sp .

We have φ(x1 + x2 ) = φ(x1 ) + φ(x2 ),

φ(λx) = λp φ(x)

q for any x1 , x2 , x ∈ HZar (X, OX ) and λ ∈ k. By 7.2.12 below, the homoq q morphism ℘ : HZar (X, OX ) → HZar (X, OX ) is surjective for each q. So we have an exact sequence ℘

q q 0 → H q (X, Z/p) → HZar (X, OX ) → HZar (X, OX ) → 0.

q If q > dim X, we have HZar (X, OX ) = 0, and hence H q (X, Z/p) = 0.

Lemma 7.2.12. Let k be an algebraically closed field of characteristic p, let V be a finite dimensional vector space over k, and let φ : V → V be a map satisfying φ(x1 + x2 ) = φ(x1 ) + φ(x2 ),

φ(λx) = λp φ(x)

for any x1 , x2 , x ∈ V and λ ∈ k. Then φ − id is surjective. Proof.

We identify V with the vector space k n . Then φ is of the form

φ(λ1 , . . . , λn ) = (a11 λp1 + · · · + an1 λpn , . . . , a1n λp1 + · · · + ann λpn ) for some aij ∈ k. So

(φ−id)(λ1 , . . . , λn ) = (a11 λp1 +· · ·+an1 λpn −λ1 , . . . , a1n λp1 +· · ·+ann λpn −λn ). Consider the k-algebra homomorphism ψ : k[x1 , . . . , xn ] → k[x1 , . . . , xn ],

xi 7→ a1i xp1 + · · · + ani xpn − xi .

It induces a k-morphism f : Spec k[x1 , . . . , xn ] → Spec k[x1 , . . . , xn ].

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The homomorphism Ωk[x1 ,...,xn ]/k → Ωk[x1 ,...,xn ]/k induced by ψ maps dxi to d(a1i xp1 + · · · + ani xpn − xi ) = −dxi . Since Ωk[x1 ,...,xn ]/k is a free k[x1 , . . . , xn ]-module with basis {dx1 , . . . , dxn }, the homomorphism of k[x1 , . . . , xn ]-modules Ωk[x1 ,...,xn ]/k ⊗ψ k[x1 , . . . , xn ] → Ωk[x1 ,...,xn ]/k induced by ψ is an isomorphism. By 2.5.3, f is etale. Hence im (f ) is open in Spec k[x1 , . . . , xn ]. The set of closed points in Spec k[x1 , . . . , xn ] can be identified with k n , and the restriction of f to this set can be identified with φ − id : k n → k n . So im (φ − id) is Zariski open in k n . But k n is an algebraic group, and im (φ − id) is an open subgroup. So we must have im (φ − id) = k n . Theorem 7.2.13. Let X be a scheme of finite type over a separably closed field k with dim X ≤ 1. For any torsion sheaf F on X, we have H q (X, F ) = 0 for all q ≥ 3. If X is affine, we have H q (X, F ) = 0 for all q ≥ 2. Proof. By 5.7.2 and 5.9.3, we may assume that k is algebraically closed and X is reduced. By 7.2.7, we have H q (X, F ) = 0 for any torsion sheaf F and any q ≥ 3. Suppose X is affine. To prove H 2 (X, F ) = 0 for a torsion sheaf F , we may assume that F is a constructible sheaf by 5.8.8 and 5.9.2. By 5.8.5 (ii), F is a constructible sheaf of Z/n-modules for some n. By 5.8.5 (i), there exists an epimorphism f! Z/n → F for some separated etale morphism f : U → X of finite type. Let K be the kernel of this epimorphism. We have H 3 (X, K ) = 0. It follows that H 2 (X, f! Z/n) → H 2 (X, F ) is onto. To prove H 2 (X, F ) = 0, it suffices to prove H 2 (X, f! Z/n) = 0. Let ηi (i = 1, . . . , m) be the generic points of X. Since f is etale, fηi : U ⊗OX OX,ηi → Spec OX,ηi are finite morphisms. By 1.10.10 (iv), for each i, there exists an open neighborhood Vi of ηi such that fVi : f −1 (Vi ) → Vi is finite. Let V = ∪i Vi and let j : V → X be the open immersion. Then the canonical morphism f! Z/n → j∗ j ∗ f! Z/n

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is an isomorphism when restricted to V . Since X − V is finite over Spec k, the q-th cohomology groups of the kernel and the cokernel of f! Z/n → j∗ j ∗ f! Z/n vanish for all q ≥ 1. So ∼ H 2 (X, j∗ j ∗ f! Z/n). H 2 (X, f! Z/n) = We are thus reduced to prove H 2 (X, j∗ j ∗ f! Z/n) = 0. We have ∼ j∗ fV ! Z/n. j∗ j ∗ f! Z/n = Since fV is finite, we have fV ! Z/n ∼ = fV ∗ Z/n by 5.5.2. It follows that j∗ j ∗ f! Z/n ∼ = (j ◦ fV )∗ Z/n.

Note that j◦fV is etale. We are thus reduced to prove that H 2 (X, f∗ Z/n) = 0 for any separated etale morphism f : U → X of finite type. By the Zariski Main Theorem 1.10.13, we have a commutative diagram i

U ,→ U f ↓ .f X such that i is a dominant open immersion and f is finite. We have H 2 (X, f∗ Z/n) ∼ = H 2 (X, f i∗ Z/n) ∼ = H 2 (U , i∗ Z/n). ∗

Note that U is affine and dim U ≤ 1. Since i is dominant, U −U is finite over Spec k. The kernel and cokernel of the canonical morphism Z/n → i∗ Z/n are supported on U − U . So we have H 2 (U , i∗ Z/n) ∼ = H 2 (U , Z/n), and we are reduced to prove H 2 (U , Z/n) = 0. We may assume that U is e be the normalization of U and let π : U e → U be the canonreduced. Let U ical morphism. Then π is finite. It induces isomorphisms above generic points of U , and hence induces an isomorphism above a dense open subset of U by 1.10.9 (ii). So the kernel and cokernel of the canonical morphism Z/n → π∗ Z/n are supported on a closed subscheme of U finite over Spec k. It follows that e , Z/n). H 2 (U , Z/n) ∼ = H 2 (U , π∗ Z/n) ∼ = H 2 (U e is a smooth affine curve. By 7.2.10, we have H 2 (U e , Z/n) = 0 if n But U 2 e is relatively prime to p = char k. By 7.2.11, we have H (U , Z/p) = 0. It e , Z/pr ) = 0 for any r ≥ 1 since Z/pr has a finite filtration follows that H 2 (U e , Z/n) = 0 with successive quotients isomorphic to Z/p. Therefore H 2 (U for all n.

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7.3

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([SGA 4] XII, XIII, [SGA 4 21 ] Arcata IV 1–4.) Consider a Cartesian square g0

X ×Y Y 0 → X f0 ↓ ↓f 0 g Y → Y.

For any sheaf F on X, applying f∗ to the canonical morphism F → g∗0 g 0∗ F , we get f∗ F → f∗ g∗0 g 0∗ F ∼ = g∗ f∗0 g 0∗ F .

From the adjointness of the functors (g ∗ , g∗ ), we get a morphism g ∗ f∗ F → f∗0 g 0∗ F .

Another way to get such a morphism is as follows: Apply g 0∗ to the canonical morphism f ∗ f∗ F → F , we get f 0∗ g ∗ f∗ F ∼ = g 0∗ f ∗ f∗ F → g 0∗ F .

From the adjointness of the functors (f 0∗ , f∗0 ), we get a morphism g ∗ f∗ F → f∗0 g 0∗ F again. At the end of this section, we show that these two ways define the same morphism g ∗ f∗ F → f∗0 g 0∗ F . For any bounded below complex K · of sheaves on X, let K · → I · and g 0∗ I · → J · be quasi-isomorphisms so that I · and J · respectively are bounded below complexes of injective sheaves on X and X ×Y Y 0 . We have morphisms g ∗ Rf∗ K · ∼ = g ∗ f∗ I · → f∗0 g 0∗ I · → f∗0 J · ∼ = Rf∗0 g 0∗ K · . In this way, we get a morphism g ∗ Rf∗ K · → Rf∗0 g 0∗ K ·

for any object K · in D+ (X). In particular, we have a morphism g ∗ Rq f∗ K · → Rq f∗0 g 0∗ K · for each q. Theorem 7.3.1 (Proper base change theorem). Let g0

X ×Y Y 0 → X f0 ↓ ↓f g Y0 → Y

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be a Cartesian square. Suppose that f is a proper morphism. Then for any torsion sheaf F on X, the canonical morphisms g ∗ Rq f∗ F → Rq f∗0 g 0∗ F

+ are isomorphisms for all q. For any object K · in Dtor (X), the canonical morphism

g ∗ Rf∗ K · → Rf∗0 g 0∗ K ·

is an isomorphism. If Rf∗ and Rf∗0 have finite cohomological dimensions, the same assertion holds for any object K · in Dtor (X). Corollary 7.3.2. Let f : X → Y be a proper morphism, let F be a torsion sheaf on X, and let s → Y be a geometric point of Y , where s is the spectrum of a separably closed field. Then (Rq f∗ F )s ∼ = H q (X ×Y s, F |X× s ) Y

for all q. Corollary 7.3.3. Let A be a strictly local ring, Y = Spec A, f : X → Y a proper morphism, and X0 the fiber of f above the closed point of Y . For any torsion sheaf F on X, the canonical homomorphisms H q (X, F ) → H q (X0 , F |X0 ) are isomorphisms for all q. 7.3.2 is a special case of 7.3.1 by taking Y 0 = s. 7.3.2 implies 7.3.3. Indeed, in the notation of 7.3.3, taking s to be the closed point of Y , we have ∼ H q (X0 , F |X ) (Rq f∗ F )s = 0

by 7.3.2. But we have (Rq f∗ F )s ∼ = H q (X, F ) by 5.9.5. So H q (X, F ) ∼ = H q (X0 , F |X0 ). 7.3.3 implies 7.3.1. Note that the first assertion of 7.3.1 implies the rest by 6.5.1 and 6.5.2. To prove the first statement, we may reduce to the case where Y = Spec A and Y 0 = Spec A0 are affine. Writing A0 as a direct limit of finitely generated A-algebras and applying 5.9.6, we may reduce to the case where Y 0 is of finite type over Y . To prove that g ∗ Rq f∗ F → Rq f∗0 g 0∗ F is an isomorphism for each q, it suffices to show that for any point y 0 in Y 0 which is a close point in the fiber g −1 (g(y 0 )), the homomorphism on stalks (g ∗ Rq f∗ F )y¯0 → (Rq f∗0 g 0∗ F )y¯0

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is an isomorphism. Indeed, if π : U → Y 0 is an etale morphism and s is a section over U of the kernel (resp. cokernel) of the morphism g ∗ Rq f∗ F → Rq f∗0 g 0∗ F , then for any point u ∈ U which is closed in the fiber (gπ)−1 (gπ(u)), there exists an etale neighborhood Wu of u ¯ over U such that s|Wu = 0. One can show that as u goes over the set of points in U which is closed in (gπ)−1 (gπ(u)), Wu form an etale covering of U . It follows that s = 0. Hence g ∗ Rq f∗ F → Rq f∗0 g 0∗ F is an isomorphism. Let us use 7.3.3 to verify (g ∗ Rq f∗ F )y¯0 → (Rq f∗0 g 0∗ F )y¯0 is an isomorphism for any q and any point y 0 in Y 0 which is closed in the fiber g −1 (g(y 0 )). Let A (resp. A0 ) be the strict henselization of OY,g(y0 ) (resp. OY 0 ,y0 ). By 5.9.5, we have (g ∗ Rq f∗ F )y¯0 ∼ = H q (X ⊗OY A, F |X⊗OY A ), = (Rq f∗ F )g(y0 ) ∼ (Rq f∗0 g 0∗ F )y¯0 ∼ = H q (X ⊗OY A0 , F |X⊗OY A0 ). By 7.3.3, we have H q (X ⊗OY A, F |X⊗OY A ) ∼ = H q (X ⊗OY k(g(y 0 )), F |X⊗O H q (X ⊗OY A0 , F |X⊗OY A0 ) ∼ = H q (X ⊗OY k(y 0 ), F |X⊗O

Y

Y

k(g(y 0 )) ),

k(y 0 ) ),

where k(g(y 0 )) (resp. k(y 0 )) is a separable closure of k(g(y 0 )) (resp. k(y 0 )). Since y 0 is closed in the fiber g −1 (g(y 0 )), k(y 0 ) is algebraic over k(y), and hence k(y 0 ) and k(y) have the same algebraic closure. It follows from 5.7.2 and 5.9.3 that we have H q (X ⊗OY k(g(y 0 )), F |X⊗O

Y

k(g(y 0 )) )

∼ = H q (X ⊗OY k(y 0 ), F |X⊗O

Y

k(y 0 ) ).

Thus (g ∗ Rq f∗ F )y¯0 ∼ = (Rq f∗0 g 0∗ F )y¯0 , and 7.3.1 holds. We will prove 7.3.3 under the extra condition that A is noetherian. It implies that 7.3.1 and 7.3.2 hold under the extra condition that Y and Y 0 are locally noetherian. This is enough for applications. Lemma 7.3.4. Let X0 → X be a morphism. For every torsion sheaf F on X, assume that the canonical homomorphism H i (X, F ) → H i (X0 , F |X0 )

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is an isomorphism for each i < n and a monomorphism for i = n. Then for every complex K · satisfying H i (K · ) = 0 for all i < 0, the canonical homomorphism H i (X, K · ) → H i (X0 , K · |X0 ) is an isomorphism for each i < n and a monomorphism for i = n. Proof. Let m ≥ 0 be an integer such that H i (K · ) = 0 for all i < m. Such m exists since H i (K · ) = 0 for all i < 0. We prove the lemma by descending induction on m. We have a biregular spectral sequence E2pq = H p (X, H q (K · )) ⇒ H p+q (X, K · ). If p + q < m, we have either p < 0 or q < m, and hence we have H p (X, H q (K · )) = 0. It follows that H i (X, K · ) = 0 for any i < m. Similarly, we have H i (X0 , K · |X0 ) = 0 for any i < m. Hence the lemma holds for those K · with H i (K · ) = 0 for all i < n + 1. Suppose that the lemma holds for those K · with H i (K · ) = 0 for all i < m + 1. Let K be a complex with H i (K · ) = 0 for all i < m. We have a distinguished triangle H m (K · )[−m] → K · → τ≥m+1 K · →, where τ≥m+1 K · is the complex · · · → 0 → K m+1 /im dm → K m+2 → · · · . Consider the commutative diagram H i−1 (X, τ≥m+1 K · ) → H i−m (X, H m (K · )) → H i (X, K · ) → ↓ ↓ ↓ H i−1 (X0 , τ≥m+1 K · |X0 ) → H i−m (X0 , H m (K · )|X0 ) → H i (X0 , K · |X0 ) → H i (X, τ≥m+1 K · ) → H i+1−m (X, H m (K · )) ↓ ↓ → H i (X0 , τ≥m+1 K · |X0 ) → H i+1−m (X0 , H m (K · )|X0 ). →

If i < n, the second vertical arrow is bijective and the fifth is injective by assumption, and the first and the fourth are bijective by the induction hypothesis. By the five lemma, the third vertical arrow is bijective. If i = n, the second vertical arrow is injective by assumption, the first is bijective and the fourth is injective by the induction hypothesis. By the five lemma, the third vertical arrow is injective. This proves our assertion.

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Lemma 7.3.5. Consider a Cartesian square i0

X00 → X 0 f0 ↓ ↓f. i

X0 → X

Assume that f is surjective, and the canonical morphism i∗ Rq f∗ F 0 → Rq f∗0 i0∗ F 0

is an isomorphism for any torsion sheaf F 0 on X 0 and any q. Then the following conditions are equivalent: (i) For any torsion sheaf F 0 on X 0 and any q, the canonical homomorphism H q (X 0 , F 0 ) → H q (X00 , i0∗ F 0 ) is an isomorphism. (ii) For any torsion sheaf F on X and any q, the canonical homomorphism H q (X, F ) → H q (X0 , i∗ F ) is an isomorphism. Proof. (i)⇒(ii) Let F → Rf∗ f ∗ F be the canonical morphism, and let F → Rf∗ f ∗ F → ∆ →

be a distinguished triangle. Since f is surjective, the canonical morphism F → f∗ f ∗ F

is injective. It follows that H i (∆) = 0 for all i < 0. We have H q (X, Rf∗ f ∗ F ) ∼ = H q (X 0 , f ∗ F ) ∼ = H q (X00 , i0∗ f ∗ F ) ∼ = H q (X0 , Rf∗0 i0∗ f ∗ F ) ∼ = H q (X0 , i∗ Rf∗ f ∗ F ).

Hence the canonical homomorphism H q (X, Rf∗ f ∗ F ) → H q (X0 , i∗ Rf∗ f ∗ F ) is an isomorphism for any q. Consider the commutative diagram 0 → H 0 (X, F ) → H 0 (X, Rf∗ f ∗ F ) → H 0 (X, ∆) ↓ ↓ ↓ 0 → H 0 (X0 , i∗ F ) → H 0 (X0 , i∗ Rf∗ f ∗ F ) → H 0 (X0 , i∗ ∆).

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The second vertical arrow is an isomorphism. It follows that the first vertical arrow is injective. Then by 7.3.4, the third vertical arrow is injective. This then implies that the first vertical arrow is an isomorphism. Suppose that we have shown that the canonical homomorphism H q (X, F ) → H q (X0 , i∗ F ) is an isomorphism for any torsion sheaf F and any q < n. Consider the commutative diagram H n−1 (X, Rf∗ f ∗ F ) → H n−1 (X, ∆) → H n (X, F ) → ↓ ↓ ↓ H n−1 (X0 , Rf∗ f ∗ F |X0 ) → H n−1 (X0 , ∆|X0 ) → H n (X0 , F |X0 ) → H n (X, Rf∗ f ∗ F ) → H n (X, ∆) ↓ ↓ n ∗ n → H (X0 , Rf∗ f F |X0 ) → H (X0 , ∆|X0 ).

→

We have shown that the first and the fourth vertical arrows are bijective. By the induction hypothesis and 7.3.4, the second vertical arrow is injective. By the five lemma, the third vertical arrow is injective. Then by 7.3.4, the fifth vertical arrow is injective and the second is bijective. By the five lemma, the third vertical arrow is bijective. (ii)⇒(i) We have H q (X 0 , F 0 ) ∼ = H q (X, Rf∗ F 0 ) ∼ = H q (X0 , i∗ Rf∗ F 0 ) ∼ = H q (X0 , Rf 0 i0∗ F 0 ) ∗

∼ = H q (X00 , i0∗ F 0 ).

Lemma 7.3.6. Suppose that A is a strictly local noetherian ring. If 7.3.3 holds for any projective morphism f , then it holds for any proper morphism f. Proof. By Chow’s lemma([Fu (2006)] 1.4.18, [EGA] II 5.6.1, [Hartshorne (1977)] Exer. II 4.10), for any proper morphism f : X → Y , there exists a surjective projective morphism g : X 0 → X such that f g is projective. We then apply 7.3.5 and the equivalence of 7.3.1 and 7.3.3. Lemma 7.3.7. Let A be a strictly local noetherian ring and let k be the residue field of A. Suppose for any nonnegative integer n and any torsion sheaf F on PnA , that the canonical homomorphisms H q (PnA , F ) → H q (Pnk , F |Pnk ) are isomorphisms for all q. Then 7.3.3 holds.

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Proof. Let

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By 7.3.6, it suffices to prove 7.3.3 for any projective morphism f . i

X → PnA f ↓ . Spec A be a commutative diagram such that i is a closed immersion, and let i0 : X0 → Pnk be the morphism induced from i by base change. For any torsion sheaf F on X, the canonical morphism (i∗ F )|Pnk → i0∗ (F |X0 ) is an isomorphism by 5.3.9. We have canonical isomorphisms ∼ H q (Pn , i∗ F ), H q (X, F ) = A

H q (X0 , F |X0 ) ∼ = H q (Pnk , i0∗ (F |X0 )).

By our assumption, we have H q (PnA , i∗ F ) ∼ = H q (Pnk , (i∗ F )|Pnk ). It follows that H q (X, F ) ∼ = H q (X0 , F |X0 ).

Lemma 7.3.8. Suppose that A is a strictly local noetherian ring. If 7.3.3 holds in the case where dim X0 ≤ 1, then it holds in general. Proof. By 7.3.7, it suffices to prove 7.3.3 for X = PnA . We use induction on n. When n = 0, this is clear. When n = 1, this follows from our assumption. Suppose that 7.3.3 holds for X = Pn−1 A . Let P be the closed subscheme of PnA ×A P1A defined by P = {([x0 : . . . : xn ], [t0 : t1 ]) ∈ PnA ×A P1A |t0 x0 + t1 x1 = 0}.

Let i : P → PnA ×A P1A ,

p1 : PnA ×A P1A → PnA ,

p2 : PnA ×A P1A → P1A

be the closed immersion and the projections, respectively. Note that p1 i : P → PnA is surjective and its fibers have dimensions ≤ 1. By our assumption, the equivalence of 7.3.3 and 7.3.1, and 7.3.5, to prove that 7.3.3 holds for X = PnA , it suffices to show that it holds for X = P . The fibers of p2 i : P → P1A are (n − 1)-dimensional projective spaces. 7.3.3 follows from 7.3.5 applied to the morphism p2 i. The conditions of 7.3.5 hold by our assumption, the induction hypothesis, and the equivalence of 7.3.1 and 7.3.3.

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Lemma 7.3.9. Suppose that A is a strictly local noetherian ring. In the notation of 7.3.3, if for any finite morphism X 0 → X and any positive integer n, the canonical homomorphism H q (X 0 , Z/n) → H q (X 0 ×X X0 , Z/n) is bijective for q = 0 and surjective for any q ≥ 1, then for any torsion sheaf F on X, the canonical homomorphism H q (X, F ) → H q (X0 , F |X0 ) is an isomorphism for any q. Proof. Suppose that F is a constructible sheaf of Z/n-modules. By 5.8.11 (ii), we can find a monomorphism F ,→ f∗ Z/n for some finite morphism f : X 0 → X. (In 5.8.11 (ii), the scheme is assumed to be of finite type over a field or over Z. To make it adapted to our case, we write A = limi Ai , −→ where Ai are subalgebras of A finitely generated over Z. By 1.10.9 and 5.9.8 (iii), we can find an Ai -scheme Xi and a constructible sheaf of Z/n-modules Fi on Xi for some large i such that the pair (X, F ) is induced from the pair (Xi , Fi ) by base change. We can apply 5.8.11 (ii) to Fi and then take base change.) Let f0 : X0 ×X X 0 → X0 be the base change of f . We have H q (X, f∗ Z/n) ∼ = H q (X 0 , Z/n), H q (X0 , (f∗ Z/n)|X0 ) ∼ = H q (X0 , f0∗ Z/n) ∼ = H q (X0 ×X X 0 , Z/n).

So the canonical homomorphism H q (X, f∗ Z/n) → H q (X0 , (f∗ Z/n)|X0 ) is bijective for q = 0 and surjective for any q > 0. Let C be the cokernel of the monomorphism F ,→ f∗ Z/n. Consider the commutative diagram H 0 (X, F ) → H 0 (X, f∗ Z/n) → H 0 (X, C ) ↓ ↓ ↓ 0 → H 0 (X0 , F |X0 ) → H 0 (X0 , (f∗ Z/n)|X0 ) → H 0 (X0 , C |X0 ). 0→

We have seen that the second vertical arrow is bijective. It follows that the first vertical arrow is injective. This is true for any constructible sheaf F of Z/n-modules. In particular, the last vertical arrow is injective. This implies that the first vertical arrow is bijective for any constructible sheaf F of Z/n-modules. By 5.8.5 (ii), 5.8.8 and 5.9.2, the canonical homomorphism H 0 (X, F ) → H 0 (X0 , F |X0 ) is bijective for any torsion sheaf F .

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Suppose that H i (X, F ) → H i (X0 , F |X0 ) is bijective for any i < q (q ≥ 1) and any torsion sheaf F . Let F be a constructible sheaf of Z/nmodules. Embed F into f∗ Z/n for some finite morphism f : X 0 → X, and embed f∗ Z/n into a flasque sheaf G of Z/n-modules. Let C1 be the cokernel of f∗ Z/n ,→ G . Consider the commutative diagram H q−1 (X, G ) → H q−1 (X, C1 ) → H q (X, f∗ Z/n) → 0 ↓ ↓ ↓ ↓ H q−1 (X0 ,G |X0 ) → H q−1 (X0 ,C1 |X0 ) →H q (X0 ,(f∗ Z/n)|X0 ) → H q (X0 ,G |X0 ). By the induction hypothesis, the first two vertical arrows are bijective. It follows that the third vertical arrow is injective. We have shown that it is surjective at the beginning. So it is bijective. Let C2 be the cokernel of F ,→ f∗ Z/n. Consider the commutative diagram H q−1 (X, f∗ Z/n) → H q−1 (X, C2 ) → H q (X, F ) → ↓ ↓ ↓ q−1 q−1 q H (X0 , (f∗ Z/n)|X0 ) → H (X0 , C2 |X0 ) → H (X0 , F |X0 ) → H q (X, f∗ Z/n) → H q (X, C2 ) ↓ ↓ → H q (X0 , (f∗ Z/n)|X0 ) → H q (X0 , C2 |X0 ).

→

By the induction hypothesis, the first two vertical arrows are bijective. We have just shown the fourth vertical arrow is bijective. So the third vertical arrow is injective. This is true for any constructible sheaf of Z/n-modules F . In particular, the last vertical arrow is injective. It then follows that the third vertical arrow is bijective for any constructible sheaf of Z/nmodules F . By 5.8.5 (ii), 5.8.8 and 5.9.2, the canonical homomorphism H q (X, F ) → H q (X0 , F |X0 ) is bijective for any torsion sheaf F . By 7.3.8 and 7.3.9, to prove 7.3.3 in the case where A is noetherian, it suffices to prove the following: Lemma 7.3.10. Let A be a strictly local noetherian ring, f : X → Spec A a proper morphism, and X0 the fiber of f over the closed point of Spec A. Suppose dim X0 ≤ 1. Then the canonical homomorphism H q (X, Z/n) → H q (X0 , Z/n) is bijective for q = 0 and surjective for any q ≥ 1.

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Proof. By 7.2.13, we have H q (X0 , Z/n) = 0 for any q ≥ 3. So it suffices to treat the cases where q = 0, 1, 2. To prove that H 0 (X, Z/n) → H 0 (X0 , Z/n) is bijective, it suffices to show that the set of connected components of X is in one-to-one correspondence with the set of connected components of X0 . Connected components of a noetherian scheme are minimal open and closed subsets. It suffices to show that the set of open and closed subsets of X is in one-to-one correspondence with the set of open and closed subsets of X0 . But the set of open and closed subsets of X (resp. X0 ) is in one-to-one correspondence with the set Idem Γ(X, OX ) (resp. Idem Γ(X0 , OX0 )) of idempotent elements in Γ(X, OX ) (resp. Γ(X0 , OX0 )). So it suffices to show that the canonical map Idem Γ(X, OX ) → Idem Γ(X0 , OX0 ) is bijective. Since X is proper over Spec A, Γ(X, OX ) is a finite A-algebra by [Fu (2006)] 2.5.3 or [EGA] III 3.2.3. By 2.8.3 (iii), the canonical map Idem Γ(X, OX ) → Idem Γ(X, OX )/mΓ(X, OX ) is bijective, where m is the maximal ideal of A. For any positive integer n, since Spec (Γ(X, OX )/mn Γ(X, OX )) has the same underlying topological space as Spec (Γ(X, OX )/mΓ(X, OX )), the canonical map Idem Γ(X, OX )/mn Γ(X, OX ) → Idem Γ(X, OX )/mΓ(X, OX ) is bijective. It follows that the canonical map Idem Γ(X, OX ) → Idem Γ(X, OX )∧ is bijective, where Γ(X, OX )∧ = lim Γ(X, OX )/mn Γ(X, OX ). ←− n

By [Fu (2006)] 2.5.6 or [EGA] III 4.1.7, we have Γ(X, OX )∧ ∼ Γ(Xn , OXn ), = lim ← − n where Xn = X ⊗A A/mn for any n. It follows that ∼ lim Idem Γ(Xn , OX ). Idem Γ(X, OX )∧ = n ←− n

But Xn has the same underlying topological space as X0 . So we have Idem Γ(Xn , OX ) ∼ = Idem Γ(X0 , OX ) n

0

and hence lim Idem Γ(Xn , OXn ) ∼ = Idem Γ(X0 , OX0 ). ←− n

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It follows that Idem Γ(X, OX ) ∼ = Idem Γ(X0 , OX0 ). This proves our assertion. To prove that H 1 (X, Z/n) → H 1 (X0 , Z/n) is surjective, we may assume that X and X0 are connected by the above discussion. By 5.7.20, we have H 1 (X, Z/n) ∼ = cont.Hom(π1 (X), Z/n), 1 H (X0 , Z/n) ∼ = cont.Hom(π1 (X0 ), Z/n). So it suffices to show that the canonical homomorphism π1 (X0 ) → π1 (X) is bijective. This follows from 7.3.13 below. Finally we prove that H 2 (X, Z/n) → H 2 (X0 , Z/n) is surjective. Let p be the characteristic of the residue field of A. By 7.2.11, we have H 2 (X0 , Z/p) = 0. For any positive integer r, Z/pr has a finite filtration with successive quotients Z/p. It follows that H 2 (X0 , Z/pr ) = 0 for any r. So to prove that H 2 (X, Z/n) → H 2 (X0 , Z/n) is surjective, we may assume that n is relatively prime to p, and prove that H 2 (X, µn,X ) → H 2 (X0 , µn,X0 ) is surjective. We have a commutative diagram ∗ ) → ··· · · · → Pic(X) → H 2 (X, µn,X ) → H 2 (X, OX et ↓ ↓ ↓ ∗ ) → ··· . · · · → Pic(X0 ) → H 2 (X0 , µn,X0 ) → H 2 (X0 , OX 0,et

So it suffices to show the canonical homomorphisms Pic(X) → Pic(X0 ),

Pic(X0 ) → H 2 (X0 , µn,X0 )

are surjective. The first homomorphism is surjective by 7.3.15 below. By 7.2.9 (i), Pic(X0red ) → H 2 (X0red , µXn,0red ) is surjective. By 5.7.2 (i), we have H 2 (X0 , µn,X0 ) ∼ = H 2 (X0red , µn,X0red ). So to prove that Pic(X0 ) → H 2 (X0 , µn,X0 ) is surjective, it suffices to prove that Pic(X0 ) → Pic(X0,red ) is surjective. This follows from 7.3.14 below. Proposition 7.3.11. Let A be a noetherian ring, I an ideal of A such that A is complete with respect to the I-adic topology, A0 = A/I, S = Spec A, S0 = Spec A0 , X a proper S-scheme, and X0 = X ×S S0 . Then the functor Y 7→ Y ×X X0 from the category of etale covering spaces of X to the category of etale covering spaces of X0 is an equivalence of categories.

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Proof.

Let Y and Y 0 be two etale covering spaces of X, and let

Sn = Spec A/I n+1 , Xn = X ×S Sn , Yn = Y ×S Sn , Yn0 = Y 0 ×S Sn . By [Fu (2006)] 2.7.20 or [EGA] III 5.4.1, we have HomX (Y, Y 0 ) ∼ HomXn (Yn , Yn0 ). = lim ←− n

By 2.3.12, the canonical maps 0 HomXn+1 (Yn+1 , Yn+1 ) → HomXn (Yn , Yn0 )

are bijective for all n. So HomX (Y, Y 0 ) ∼ = HomX0 (Y0 , Y00 ) and the functor Y 7→ Y ×X X0 is fully faithful. Let f0 : Y0 → X0 be an etale covering space. By 2.3.12, there exist etale morphisms fn : Yn → Xn such that Yn ∼ = Yn+1 ×Xn+1 Xn for all n. Since X0 and Xn have the same underlying topological spaces and f0 is proper, fn is also proper. Since fn is etale, it is quasi-finite. By [Fu (2006)] 2.5.12 or [EGA] III 4.4.2, fn is finite. Let Bn = fn∗ OXn . Then Bn is a coherent OXn -algebra and Yn = Spec Bn . We have Bn+1 ⊗OXn+1 OXn ∼ = Bn . By Grothendieck Existence Theorem ([Fu (2006)] 2.7.11 or [EGA] III 5.1.4), there exists a coherent OX -module B such that Bn ∼ = B ⊗OX OXn . Defining an OX -algebra structure on B is equivalent to defining a morphism B ⊗ OX B → B which makes various diagrams expressing the associative law, the commutative law and the distributive law to commute, and that these diagrams involve only tensor products of B. Using [Fu (2006)] 1.5.19 and 2.7.9, or [EGA] I 10.11.4 and III 5.1.3, one can show that B has an OX -algebra structure such that Bn ∼ = B ⊗OX OXn are isomorphisms of OXn -algebras. For any closed point x in X, since X is proper over S, x is above the closed point of S. Since f0 : Spec B0 → X0 is etale, B0 ⊗OX0 k(x) is a direct product of finite separable extensions of k(x). We have B ⊗OX k(x) ∼ = B0 ⊗OX0 k(x).

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We will prove in a moment that B is locally free as an OX -module. So the morphism Spec B → X is etale. It is an etale covering space of X inducing f0 : Y0 → X0 . So the functor Y 7→ Y ×X X0 is essentially surjective. Finally let us prove that B is locally free as an OX -module. It suffices to show that the functor G 7→ H omOX (B, G ) b be the formal is exact on the category of coherent OX -modules. Let X ∼ completion of X with respect to I OX . For any coherent OX -module G , let Gb be the formal completion of G with respect to I ∼ OX . By [Fu (2006)] 1.5.13 (i) and 2.7.9, or [EGA] I 10.8.8 (i) and III 5.1.3, the functor G 7→ Gb is exact and faithful. So to prove that the functor G 7→ H omOX (B, G ) is exact, it suffices to show that the functor G 7→ H omOX (B, G )∧

is exact. By [Fu (2006)] 1.5.13 (iii) and 1.5.19, or [EGA] I 10.8.10 and 10.11.7, we have b Gb) H omO (B, G )∧ ∼ = H omO (B, X

c X

∼ H omOXn (Bn , Gn ), = lim ←− n

where Gn is the inverse image of G on Xn . By our construction, each Bn is locally free. Moreover, we can find an open covering {Uλ } of X0 such that Bn |Uλ is free for each λ. Here we regard each Uλ as an open subset of Xn . Using this fact, one can prove that the functor G 7→ lim H omOXn (Bn , Gn ) ←− n

is exact. This proves our assertion.

Theorem 7.3.12 (Artin’s Approximation Theorem). Let A be the henselization at a prime ideal of a finitely generated algebra over a field or over an excellent discrete valuation ring, let m be the maximal ideal of b be the m-adic completion of A, and let A, let A f1 (Y1 , . . . , Yn ), . . . , fm (Y1 , . . . , Yn ) ∈ A[Y1 , . . . , Yn ]. b such that If there exist yˆ1 , . . . , yˆn ∈ A f1 (ˆ y1 , . . . , yˆn ) = · · · = fm (ˆ y1 , . . . , yˆn ) = 0,

then for any given positive integer N , there exist y1 , . . . , yn ∈ A such that b yi ≡ yˆi mod mN A, f1 (y1 , . . . , yn ) = · · · = fm (y1 , . . . , yn ) = 0.

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For a proof of Artin’s approximation theorem, confer [Artin (1969)] or [Bosch, L¨ utkebohmert and Raynaud (1990)] 3.6. Lemma 7.3.13. Let A be a noetherian henselian local ring, f : X → Spec A a proper morphism, and X0 the fiber of f over the closed point of Spec A. The functor Y 7→ Y ×X X0 from the category of etale covering spaces of X to the category of etale covering spaces of X0 is an equivalence of categories. Proof. We can write A = limλ Bλ , where each Bλ is a henselization at a −→ prime ideal of an algebra finitely generated over Z. By 1.10.9 and 1.10.10 (xi), we may assume that f : X → Spec A can be descended down to an inverse system of proper morphisms fλ : Xλ → Spec Bλ . For each λ, let Xλ0 be the fiber of fλ over the closed point of Spec Bλ . By 2.3.7 and 1.10.10 (iv), any etale covering space of X (resp. X0 ) can be descended down to an etale covering space of Xλ (resp. Xλ0 ) for sufficiently large λ. Moreover, if Y and Y 0 are two etale covering spaces of X which can be descended down to etale covering spaces Yλ and Yλ0 of Xλ , respectively, we have HomX (Y, Y 0 ) ∼ HomXµ (Yµ , Yµ0 ), = lim −→ µ≥λ

HomX0 (Y0 , Y00 )

0 ∼ HomXµ0 (Yµ0 , Yµ0 ), = lim −→ µ≥λ

where Yµ (resp. Yµ0 ) are induced by Yλ (resp. Yλ0 ) by the base changes 0 Xµ → Xλ , Y0 and Y00 (resp. Yµ0 and Yµ0 ) are the fibers of Y and Y 0 (resp. 0 Yµ and Yµ ) over the closed point of Spec A (resp. Spec Bµ ). It follows that to prove the lemma for A, it suffices to prove the lemma for each Bλ . We may thus assume that A is the henselization at a prime ideal of a finitely generated algebra over Z so that Artin’s Approximation Theorem is applicable. Let Y and Y 0 be two etale covering spaces of X. Let us prove that HomX (Y, Y 0 ) → HomX0 (Y0 , Y00 )

is injective. Supppose g1 , g2 ∈ HomX (Y, Y 0 ) induce the same element in b the mHomX0 (Y0 , Y00 ). Denote by m the maximal ideal of A, and by A adic completion of A. By 7.3.11, g1 and g2 induce the same element in 0 HomX (Y , Y ), where b X = X ⊗A A,

b Y = Y ⊗A A,

0

b Y = Y 0 ⊗A A.

b = lim Aα , where {Aα } is the family of subalgebras of A b Write A −→α finitely generated over A. Then g1 and g2 induce the same element in

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HomXα (Yα , Yα0 ) for a sufficiently large Aα , where Xα = X ⊗A Aα ,

Yα = Y ⊗A Aα ,

Yα0 = Y 0 ⊗A Aα .

Fix an isomorphism of A-algebras A[Y1 , . . . , Yn ]/(f1 , . . . , fm ) ∼ = Aα b induces an for some f1 , . . . , fm ∈ A[Y1 , . . . , Yn ]. The inclusion Aα ,→ A A-homomorphism b A[Y1 , . . . , Yn ]/(f1 , . . . , fm ) → A.

b be the images of Y1 , . . . , Yn under this homomorphism, Let yˆ1 , . . . , yˆn ∈ A respectively. Then f1 (ˆ y1 , . . . , yˆn ) = · · · = fm (ˆ y1 , . . . , yˆn ) = 0.

By Artin’s Approximation Theorem, there exist y1 , . . . , yn ∈ A such that b yi ≡ yˆi mod mA,

f1 (y1 , . . . , yn ) = · · · = fm (y1 , . . . , yn ) = 0. We have a homomorphism of A-algebras A[Y1 , . . . , Yn ]/(f1 , . . . , fm ) → A which maps Yi to yi . It induces a homomorphism of A-algebras φ : Aα → A. We have X∼ = Xα ⊗Aα ,φ A,

Y ∼ = Yα ⊗Aα ,φ A,

Y0 ∼ = Yα0 ⊗Aα ,φ A.

Since g1 and g2 induce the same element in HomXα (Yα , Yα0 ), it follows that g1 and g2 are the same in HomX (Y, Y 0 ). Next we show that HomX (Y, Y 0 ) → HomX0 (Y0 , Y00 ) is surjective. Let 0 g0 ∈ HomX0 (Y0 , Y00 ). By 7.3.11, there exists g¯ ∈ HomX (Y , Y ) inducing g0 . By 1.10.9, g¯ can be descended down to a morphism gα ∈ HomXα (Yα , Yα0 ) for a sufficiently large Aα . Let g ∈ HomX (Y, Y 0 ) be the morphism induced by gα by the base change φ : Aα → A. Then g is the preimage of g0 . Finally let Y0 be an etale covering space of X0 . By 7.3.11, it is induced by an etale covering space Y of X. By 1.10.9, 1.10.10 (iv) and 2.3.7, we may descend Y down to an etale covering space Yα of Xα for a sufficiently large Aα . Let Y be the etale covering space of X induced by Yα by base change φ : Aα → A. Then Y0 is induced by Y .

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Lemma 7.3.14. Let X be a noetherian scheme of dimension ≤ 1, and let I be a coherent OX -ideal satisfying I n = 0 for some nonnegative integer n. Then the canonical homomorphism Pic (X) → Pic (Spec (OX /I )) is surjective. Proof. First consider the case where I 2 = 0. We have a short exact sequence 1+id

∗ 0 → I → OX → (OX /I )∗ → 0,

where ∗ 1 + id : I → OX

is defined by (1 + id)(s) = 1 + s for any section s of I . This is a morphism of sheaves of abelian groups since I 2 = 0. The above short exact sequence induces a long exact sequence ∗ 1 2 1 (X, I ) → · · · . (X, (OX /I )∗ ) → HZar · · · → HZar (X, OX ) → HZar

We have 1 ∗ ∼ HZar (X, OX ) = Pic (X),

1 (X, (OX /I )∗ ) ∼ HZar = Pic (Spec (OX /I )).

2 Since dim X ≤ 1, we have HZar (X, I ) = 0 by [Hartshorne (1977)] III 2.7. So Pic (X) → Pic (Spec (OX /I )) is surjective. In general, Pic (Spec (OX /I k+1 )) → Pic (Spec (OX /I k )) is surjective for all k. It follows that Pic (X) → Pic (Spec (OX /I )) is surjective if I n = 0 for some nonnegative integer n.

Lemma 7.3.15. Let A be a noetherian henselian local ring, f : X → Spec A a proper morphism, and X0 the fiber of f over the closed point of Spec A. Suppose dim X0 ≤ 1. Then the canonical homomorphism Pic(X) → Pic(X0 ) is surjective. Proof. As in the proof of 7.3.13, we may assume that A is the henselization at a prime ideal of an algebra finitely generated over Z so that Artin’s Approximation Theorem is applicable. Let m be the maximal ideal of A, b be the m-adic completion of A, let Xn = X ⊗A A/mn+1 , and let let A b Given an invertible OX0 -module L0 , by 7.3.14, for each n, X = X ⊗A A. we may find an invertible OXn -module Ln such that its inverse image in Xn+1 is isomorphic to Ln+1 . By Grothendieck’s Existence Theorem ([Fu (2006)] 2.7.11 or [EGA] III 5.1.4), there exists a coherent OX -module L such that its inverse image on Xn is isomorphic to Ln for each n. As in the proof of 7.3.11, one can show the functor H omOX (L , −) is exact on

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the category of coherent OX -modules. So L is an invertible OX -module. b = lim Aα , where {Aα } is the family of As in the proof of 7.3.13, write A −→α b finitely generated over A. Let Xα = X ⊗A Aα . By 1.10.2, subalgebras of A

we may descend L to an invertible OXα -module Lα for a sufficiently large Aα . Let L be the invertible OX -module induced by Lα by the base change φ : Aα → A constructed in the proof of 7.3.13. Then L is a preimage of L0 under the canonical homomorphism Pic(X) → Pic(X0 ). Finally, we prove that the two ways of defining the base change morphism at the beginning of this section are the same. Proposition 7.3.16. Let g0

X0 → X f 0↓ ↓f g Y0 → Y be a commutative diagram, and let F be a sheaf on X. Then the morphisms g ∗ f∗ F → f∗0 g 0∗ F defined by the following two ways are the same: (i) Take the composite f∗ F

f∗ (adj)

→

f∗ g∗0 g 0∗ F ∼ = g∗ f∗0 g 0∗ F ,

and define g ∗ f∗ F → f∗0 g 0∗ F to be the morphism induced by the composite by adjunction. (ii) Take the composite f 0∗ g ∗ f∗ F ∼ = g 0∗ f ∗ f∗ F

g0∗ (adj)

→

g 0∗ F ,

and define g ∗ f∗ F → f∗0 g 0∗ F to be the morphism induced by the composite by adjunction. Proof. Define a category C as follows: Objects in C are pairs (Z, G ) such that Z is a scheme and G is a sheaf on Z. Given two objects (Z1 , G1 ) and (Z2 , G2 ) in C , a morphism (Z1 , G1 ) → (Z2 , G2 ) in C is a pair (h, φ) such that h : Z1 → Z2 is a morphism of schemes and φ : G2 → h∗ G1 is a morphism of sheaves. We define composites of morphisms in C in the obvious way. Note that we can also define a morphism (Z1 , G1 ) → (Z2 , G2 ) in C to be a pair (h, ψ) such that h : Z1 → Z2 is a morphism of schemes and ψ : h∗ G2 → G1 is a morphism of sheaves. Given a morphism h : Z1 → Z2 of schemes and sheaves G1 on Z1 and G2 on Z2 , denote by Homh ((Z1 , G1 ), (Z2 , G2 )) the set

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of morphisms (h, φ) : (Z1 , G1 ) → (Z2 , G2 ) in C whose first components are h. Given two morphism h

h

Z1 →1 Z2 →2 Z3 of schemes and sheaves Gi (i = 1, 2, 3) on Zi , we have one-to-one correspondences Homh1 ((Z1 , G1 ), (Z2 , h∗2 G3 )) ∼ = Homh2 h1 ((Z1 , G1 ), (Z3 , G3 )), ∼ Homh h ((Z1 , G1 ), (Z3 , G3 )). Homh2 ((Z2 , h1∗ G1 ), (Z3 , G3 )) = 2 1 In the notation of the proposition, we have one-to-one correspondences Hom(g ∗ f∗ F , f∗0 g 0∗ F ) ∼ = HomidY 0 ((Y 0 , f∗0 g 0∗ F ), (Y 0 , g ∗ f∗ F )) ∼ = Homg ((Y 0 , f∗0 g 0∗ F ), (Y, f∗ F )) ∼ = Homgf 0 ((X 0 , g 0∗ F ), (Y, f∗ F ))

∼ = Homf g0 ((X 0 , g 0∗ F ), (Y, f∗ F )). Let α be the morphism

(g 0 , idg0∗ F ) ∈ Homg0 ((X 0 , g 0∗ F ), (X, F )), and let β be the morphism (f, idf∗ F ) ∈ Homf ((X, F ), (Y, f∗ F )). Let us prove that the two morphisms g ∗ f∗ F → f∗0 g 0∗ F defined in the proposition both correspond to the morphism βα ∈ Homf g0 ((X 0 , g 0∗ F ), (Y, f∗ F )) through the above one-to-one correspondences. For any scheme Z and any morphism φ : G1 → G2 of sheaves on Z, denote the morphism (id, φ) : (Z, G2 ) → (Z, G1 ) in C by φ? . Let u be the composite f∗ F

f∗ (adj)

→

f∗ g∗0 g 0∗ F ∼ = g∗ f∗0 g 0∗ F ,

and let θ1 : g ∗ f∗ F → f∗0 g 0∗ F be the morphism defined in (i) of the proposition. We have a commutative diagram (Y 0 , f∗0 g 0∗ F ) → (Y, g∗ f∗0 g 0∗ F ) ∼ = (Y, f∗ g∗0 g 0∗ F ) ← (X, g∗0 g 0∗ F ) ← (X 0 , g 0∗ F ) ? ? θ1 ↓ u ↓ ↓ f∗ (adj)? ↓ adj? ↓α (Y 0 , g ∗ f∗ F ) →

(Y, f∗ F )

=

(Y, f∗ F )

β

←

(X, F )

= (X, F ),

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where the horizontal arrows are the canonical morphisms. Note that the composite of canonical morphisms (X 0 , g 0∗ F ) → (X, g∗0 g 0∗ F ) → (Y, f∗ g∗0 g 0∗ F ) ∼ = (Y, g∗ f∗0 g 0∗ F ) coincides with the composite of canonical morphisms (X 0 , g 0∗ F ) → (Y 0 , f∗0 g 0∗ F ) → (Y, g∗ f∗0 g 0∗ F ). So we have a commutative diagram (Y 0 , f∗0 g 0∗ F ) θ1? ↓

← β

(X 0 , g 0∗ F ) ↓α

(Y 0 , g ∗ f∗ F ) → (Y, f∗ F ) ← (X, F ).

This shows that θ1 corresponds to βα. Similarly, let v be the composite f 0∗ g ∗ f∗ F ∼ = g 0∗ f ∗ f∗ F

g0∗ (adj)

→

g 0∗ F ,

and let θ2 : g ∗ f∗ F → f∗0 g 0∗ F be the morphism defined in (ii) of the proposition. We have a commutative diagram α

(Y 0 , f∗0 g 0∗ F ) ← (X 0 , g 0∗ F ) = (X 0 , g 0∗ F ) → (X, F ) = (X, F ) θ2? ↓ v? ↓ ↓ g 0∗ (adj)? ↓ adj? ↓β 0 ∗ 0 0∗ ∗ 0 0∗ ∗ ∼ (Y , g f∗ F ) ← (X , f g f∗ F ) = (X , g f f∗ F ) → (X, f ∗ f∗ F ) → (Y, f∗ F ), where the horizontal arrows are the canonical morphisms. The composite of canonical morphisms (X 0 , f 0∗ g ∗ f∗ F ) ∼ = (X 0 , g 0∗ f ∗ f∗ F ) → (X, f ∗ f∗ F ) → (Y, f∗ F ) coincides with the composite of canonical morphisms (X 0 , f 0∗ g ∗ f∗ F ) → (Y 0 , g ∗ f∗ F ) → (Y, f∗ F ). So we have a commutative diagram α

(Y 0 , f∗0 g 0∗ F ) ← (X 0 , g 0∗ F ) → (X, F ) θ2? ↓ ↓β 0 ∗ (Y , g f∗ F ) → (Y, f∗ F ). This shows that θ2 corresponds to βα.

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Cohomology with Proper Support

([SGA 4] XVII 3–6, [SGA 4 21 ] Arcata IV 5, 6.) Throughout this section, we fix a scheme S. Let X and Y be S-schemes. An S-morphism f : X → Y is called S-compactifiable if Y is quasi-compact and quasi-separated and there exists a proper S-scheme P such that f is the composite of a separated quasi-finite morphism X → Y ×S P and the projection p1 : Y ×S P → Y . Note that f is then a quasi-compact separated morphism, and X is a quasi-compact quasi-separated scheme. We say that an S-scheme X is compactifiable if the structure morphism X → S is Scompactifiable. p2 X → Y ×S P → P f & p1 ↓ ↓ Y →S A theorem of Nagata says that any separated morphism of finite type between noetherian schemes is compactifiable. Proposition 7.4.1. (i) S-compactifiable morphisms are separated and of finite type. (ii) Let f : X → Y be a separated quasi-finite S-morphism such that Y is quasi-compact and quasi-separated. Then f is S-compactifiable. (iii) Let f : X → Y and g : Y → Z be two S-compactifiable morphisms. Then gf is S-compactifiable. (iv) Let f : X → Y be an S-compactifiable morphism. For any morphism Y 0 → Y such that Y 0 is quasi-compact and quasi-separated, the base change f 0 : X ×Y Y 0 → Y 0 is S-compactifiable. (v) Let f : X → Y and g : Y → Z be two S-morphisms. Suppose g is quasi-compact and quasi-separated and gf is S-compactifiable. Then f is S-compactifiable. Proof. (i), (ii) and (iv) are clear. (v) follows from (iii) and (iv). (iii) We can find proper S-schemes P and Q, and separated quasi-finite morphisms X → Y ×S P and Y → Z ×S Q, so that their composites with the projections to the first components are f and g, respectively. Consider the commutative diagram X → Y ×S P → Z ×S P ×S Q & ↓ ↓ Y → Z ×S Q & ↓ Z.

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Note that the square in the above diagram is Cartesian. All horizontal arrows are separated and quasi-finite, and all vertical arrows are projections. As P ×S Q is proper over S, gf is S-compactifiable. Let f : X → Y be an S-compactifiable morphism. Then we can find a proper S-scheme P and a separated quasi-finite morphism g : X → Y ×S P such that f = p1 g. By the Zariski Main Theorem in [EGA] IV 18.12.13, (we treat the noetherian case in 1.10.13), we can find a finite morphism g¯ and an open immersion j such that g = g¯j. By 7.4.1 (ii), g¯ is S-compactifiable, and by 7.4.1 (iii), p1 g¯ is S-compactifiable. We have f = (p1 g¯)j. So any S-compactifiable morphism is the composite of an open immersion and a proper S-compactifiable morphism. f Let X → Y be a compactifiable S-morphism. A compactification of f is a factorization of f as a composite j

f¯

X ,→ X → Y

such that j is an open immersion, and f¯ is a proper S-compactifiable morphism. Note that j is necessarily an S-compactifiable morphism. A morphism from the above compactification of f to another compactification j0

0 f¯0

X ,→ X → Y

0 is a morphism φ : X → X such that φj = j 0 and f¯0 φ = f¯. If such a morphism exists, we say the first compactification dominates the second one. f g Let X → Y → Z be S-compactifiable morphisms. A compactification of (f, g) is a commutative diagram

X ,→ X ,→ X & ↓ ↓ Y ,→ Y & ↓ Z

0

such that horizontal arrows are open immersions, and vertical arrows are proper S-compactifiable morphisms. Note that all arrows in the above diagram are necessarily S-compactifiable morphisms. We leave it for the reader to define morphisms between compactifications of (f, g). f

f

g

Proposition 7.4.2. Let X → Y (resp. X → Y → Z) be S-compactifiable morphism(s). (i) There exists a compactification for f (resp. (f, g)).

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(ii) Given two compactifications of f , there exists a compactification dominating both of them. (iii) Given two compactifications ji

0

00

f¯i

X ,→ X i → Y

(i = 1, 2)

of f , let φ , φ : X 1 → X 2 be two morphisms between these compactifications. There exists a compactification X ,→ X → Y

and a morphism φ : X → X 1 of compactifications such that φ0 ◦ φ = φ00 ◦ φ. (iv) Given two compactifications of (f, g), there exists a compactification dominating both of them. Proof. (i) We have seen that f has a compactification. Choose compactifications X ,→ X → Y and Y ,→ Y → Z

of f and g, respectively. By 7.4.1 (iii), the composite X → Y ,→ Y is 0 S-compactifiable. Choose a compactification X ,→ X → Y for this composite. We then get a commutative diagram 0

X ,→ X ,→ X & ↓ ↓ Y ,→ Y & ↓ Z, which is a compactification of (f, g). (ii) Given two compactifications X ,→ X i → Y

(i = 1, 2)

for f , the morphism X 1 ×Y X 2 → Y is S-compactifiable by 7.4.1 (iii) and (iv). The morphism X → X 1 ×Y X 2 defined by X ,→ X i (i = 1, 2) is an immersion. It is S-compactifiable by 7.4.1 (v). So we can factorize it as the composite of an open immersion X ,→ X and a closed immersion X → X 1 ×Y X 2 . The composite X → X 1 ×Y X 2 → Y is S-compactifiable by 7.4.1 (ii) and (iii), and it is proper. So X ,→ X → Y is a compactification of f , and it dominates the given two compactifications. (iii) Define K by the Cartesian square p2

K → X1 p1 ↓ ↓ Γφ00 Γφ0

X 1 → X 1 ×Y X 2 ,

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where Γφ0 , Γφ00 : X 1 → X 1 ×Y X 2 are the graphs of φ0 and φ00 , respectively. Let πi : X 1 ×Y X 2 → X i (i = 1, 2) be the projections. We have p1 = id ◦ p1 = π1 Γφ0 p1 = π1 Γφ00 p2 = id ◦ p2 = p2 . Denote p1 = p2 by p. Note that p is a closed immersion. One can show that the sequence φ0

p

K → X1 ⇒ X2 φ00

is exact, that is, φ0 p = φ00 p, and for any morphism ψ : K 0 → X 1 with the property φ0 ψ = φ00 ψ, there exists a unique morphism ψ 0 : K 0 → K such that pψ 0 = ψ. We have φ0 j1 = j2 = φ00 j1 . So there exists a morphism j : X → K such that pj = j1 . j is necessarily an open immersion. The morphism f¯1 p : K → Y is S-compactifiable by 7.4.1 (ii) and (iii), and it is proper. So f¯1 p

j

X ,→ K → Y is a compactification of f , and p is a morphism between compactifications such that φ0 φ = φ00 φ. X = ↓

j

X = X ↓ ↓ j2

j1

φ0

p

K → X1 ⇒ X2 φ00

f1

↓ ↓ f2 Y = Y

(iv) Given two compactifications 0

X ,→ X i ,→ X i & ↓ ↓ Y ,→ Y i & ↓ Z

(i = 1, 2)

of (f, g), find a compactification X ,→ X → Y

(resp. Y ,→ Y → Z)

X ,→ X i → Y

(resp. Y ,→ Y i → Z)

dominating

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for i = 1, 2. For each i, we have a commutative diagram 0

X i ,→ X i % ↓ ↓ X Y ,→ Y i . ↓ % % Y ,→ Y Let 0

X → X i ×Y i Y be the Y i -morphism with components given by the composites 0

X → X i → X i,

X →Y →Y.

It is S-compactifiable by 7.4.1 (v). Let 0

X ,→ T i → X i ×Y i Y be a compactification. Then the diagram X ,→ T i ↓ ↓ Y ,→ Y dominates the diagram 0

X i ,→ X i ↓ ↓ Y ,→ Y i . The morphism T i → Y is S-compactifiable and proper. It follows that X ,→ T i → Y is a compactification of the composite X → Y → Y . Choose a compactification 0

X ,→ X → Y of X → Y → Y dominating X ,→ T i → Y for i = 1, 2. Then the diagram X ,→ X ,→ X & ↓ ↓ Y ,→ Y & ↓ Z

0

is a compactification of (f, g) dominating the given compactifications.

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Lemma 7.4.3. Consider a commutative diagram k

X ,→ X f ↓ ↓ f¯ j

Y ,→ Y ,

where k and j are open immersions, and f and f¯ are proper morphisms. + Then for any K ∈ ob Dtor (X), we have a canonical isomorphism ∼ = j! Rf∗ K → Rf¯∗ k! K.

Proof.

Fix notation by the commutative diagram l

j0

i0

j

i

X ,→ X ×Y Y ,→ X ← X ×Y (Y − Y ) f & f¯0 ↓ ↓ f¯ ↓ f¯00 Y ,→ Y ← Y − Y,

where the squares are Cartesian, i : Y − Y → Y is a closed immersion, and l is the morphism such that j 0 ι = k and f¯0 l = f . Since j is an open immersion, we have a canonical isomorphism ∼ = j ∗ Rf¯∗ k! K → Rf¯∗0 l! K.

Since f and f¯0 are proper, l is also proper. Since j 0 and k are open immersions, l is also an open immersion. It follows that l is an open and closed immersion and hence l! K ∼ = Rl∗ K. So we have Rf¯∗0 l! K ∼ = Rf¯∗0 Rl∗ K ∼ = Rf∗ K. We thus have an isomorphism j ∗ Rf¯∗ k! K ∼ = Rf∗ K. The inverse of this isomorphism induces a morphism j! Rf∗ K → Rf¯∗ k! K, and its restriction to Y is an isomorphism. To show that it is an isomorphism, it suffices to show i∗ Rf¯∗ k! K = 0. We have i0∗ k! K = 0, and by the proper base change theorem 7.3.1, we have i∗ Rf¯∗ k! K ∼ = Rf¯∗00 i0∗ k! K = 0.

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Let f : X → Y be an S-compactifiable morphism, and let f¯i

ji

X ,→ X i → Y

(i = 1, 2)

be two compactifications of f . Choose a compactification j3

f¯3

X ,→ X 3 → Y dominating both of them, and let φi : X 3 → X i (i = 1, 2) be morphisms making the following diagrams commute: j3

f¯3

ji

f¯i

X ,→ X 3 → Y k φi ↓ k

(i = 1, 2)

X ,→ X i → Y. Then φi are proper, and by 7.4.3, we have isomorphisms ∼ =

ji! K → Rφi∗ j3! K

+ for any K ∈ ob Dtor (X). So we have isomorphisms

∼ = Rf¯i∗ ji! K → Rf¯i∗ Rφi∗ j3! K ∼ = Rf¯3∗ j3! K.

We thus have an isomorphism Rf¯1∗ j1! K ∼ = Rf¯2∗ j2! K. This isomorphism is independent of the choice of the third compactification of f dominating the first two compactifications. Indeed, given a fourth compactification dominating the first two, by 7.4.2 (ii) and (iii), we can find a fifth compactification dominating the third and the fourth such that for each i = 1, 2, the composite of the morphism from the fifth to the third and the morphism from the third to the i-th coincides with the composite of the morphism from the fifth to the fourth and the morphism from the fourth to the i-th. It is clear that the isomorphisms Rf¯1∗ j1! K ∼ = Rf¯2∗ j2! K constructed from both the third and the fourth compactifications coincide with the isomorphism constructed from the fifth compactification. So the isomorphism Rf¯1∗ j1! K ∼ = Rf¯2∗ j2! K is independent of the choice of the compactification dominating the first two compactifications. For any S-compactifiable morphism f : X → Y , choose a compactification j

f¯

X ,→ X → Y.

+ For any K ∈ ob Dtor (X), we define

Rf! K = Rf¯∗ j! K.

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By the above discussion, the isomorphic class of the functor Rf! is independent of the choice of the compactification. Define Rq f! K = H q (Rf! K). Let F be a torsion sheaf on X, we define f! F = R0 f! F = f¯∗ j! F . Note that f! coincides with the functor defined in 5.5. Suppose that f : X → Y is a separated quasi-finite S-morphism such that Y is quasi-compact and quasi-separated. By the Zariski Main Theorem, we have a compactification f¯

j

X ,→ X → Y

such that f¯ is finite. We then have

Rf! = Rf¯∗ j! = f¯∗ j! = f! . Suppose that S = Spec k is the spectrum of a separably closed field k. Let f : X → S be an S-compactifiable morphism, and let j

f¯

X ,→ X → Spec k + be a compactification. For any K ∈ ob Dtor (X), we define

RΓc (X, K) = Γ(Spec k, Rf! K) ∼ = RΓ(X, j! K), q q ∼ H (X, K) = H (RΓc (X, K)) = H q (X, j! K). c

For any torsion sheaf F on X, we define Γc (X, F ) = Hc0 (X, F ) ∼ = Γ(X, j! F ). Note that Γc (X, F ) consists of those sections in Γ(X, F ) with support proper over Spec k. In general, Rf! is not the right derived functor of f! . For example, in the case where S = Spec k is the spectrum of a separably closed field k and X is a smooth affine algebraic curve over k, we have M Γc (X, F ) ∼ Γx (X, F ), = x∈|X|

where |X| is the set of Zariski closed points in X. It follows that the right L derived functor of Γc (X, −) is isomorphic to x∈|X| RΓx (X, −). Later we L will show Hx1 (X, Z/n) ∼ = Z/n. Thus x∈|X| Hx1 (X, Z/n) is infinite. But we 1 will see that Hc (X, Z/n) is finite.

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Theorem 7.4.4. (i) Consider a Cartesian diagram g0

X ×Y Y 0 → X f0 ↓ ↓f g Y 0 → Y.

Suppose that f is S-compactifiable, and Y 0 is quasi-compact and quasi+ separated. Then for any K ∈ ob Dtor (X), we have a canonical isomorphism ∼ =

f

g

g ∗ Rf! K → Rf!0 g 0∗ K.

(ii) Let X → Y → Z be two S-compactifiable morphisms. For any + K ∈ ob Dtor (X), we have a canonical isomorphism ∼ =

Rg! Rf! K → R(gf )! K, and we have a biregular spectral sequence E2pq = Rp g! Rq f! K ⇒ Rp+q (gf )! K. (iii) Let f : X → Y be an S-compactifiable morphism, j : U ,→ X an S-compactifiable open immersion, and i : A → X a closed immersion with + A = X − U . For any K ∈ ob Dtor (X), we have a distinguished triangle R(f j)! j ∗ K → Rf! K → R(f i)! i∗ K →

and a long exact sequence · · · → Rq (f j)! j ∗ K → Rq f! K → Rq (f i)! i∗ K → Rp+1 (f j)! j ∗ K → · · · . Proof. (i) Let j

f¯

X ,→ X → Y be a compactification of f . Fix notation by the following commutative diagram g0

X ×Y Y 0 → X j0 ↓ ↓j g ¯

We have

X ×Y Y 0 → X f¯0 ↓ ↓ f¯ g Y 0 → Y.

j!0 g 0∗ K ∼ = g¯∗ j! K,

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and by the proper base change theorem 7.3.1, we have g ∗ Rf¯∗ j! K ∼ = Rf¯0 g¯∗ j! K. ∗

It follows that g ∗ Rf¯∗ j! K ∼ = Rf¯∗0 j!0 g 0∗ K, that is, g ∗ Rf! K ∼ = Rf!0 g 0∗ K. One can verify that this isomorphism is independent of the choice of the compactification of f . (ii) Let X

j2

j1

0

,→ X ,→ X f & f¯ ↓ ↓ f¯0 Y

k

,→ Y g & ↓ g ¯ Z

be a compactification of (f, g). By 7.4.3, we have k! Rf¯∗ j1! K ∼ = Rf¯0 j2! j1! K. ∗

So we have Rg! Rf! K ∼ g∗ k! Rf¯∗ j1! K ∼ g∗ Rf¯∗0 j2! j1! K ∼ gf¯0 )∗ (j2 j1 )! K ∼ = R¯ = R¯ = R(¯ = R(gf )! K, ∼ that is, Rg! Rf! K = R(gf )! K. This isomorphism is independent of the choice of the compactification of (f, g). + For any L ∈ ob Dtor (Y ), we have a biregular spectral sequence E2pq = Rp g¯∗ (H q (k! L)) ⇒ Rp+q g¯∗ k! L.

Taking L = Rf! K, we get the spectral sequence E2pq = Rp g! Rq f! K ⇒ Rp+q (gf )! K. (iii) By 5.4.2 (iv), we have a distinguished triangle j! j ∗ K → K → i∗ i∗ K →, and hence a distinguished triangle Rf! j! j ∗ K → Rf! K → Rf! i∗ i∗ K → . ∼ R(f j)! j ∗ K and Rf! i∗ i∗ K ∼ We have Rf! j! j ∗ K = = R(f i)! i∗ K. So we have a distinguished triangle R(f j)! j ∗ K → Rf! K → R(f i)! i∗ K → .

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Theorem 7.4.5. Let f : X → Y be an S-compactifiable morphism and let n be the supremum of dimensions of fibers of f . Then for any torsion sheaf F on X, we have Rq f! F = 0 for all q > 2n. Proof.

By 7.4.4 (i), for any y ∈ Y , we have

(Rq f! F )y¯ ∼ = Hcq (X ⊗OY k(y), F |X⊗O

Y

k(y) ),

where k(y) is a separable closure of the residue field k(y). So it suffices to prove that for any separably closed field k and any S-compactifiable morphism X → Spec k, we have Hcq (X, F ) = 0 for any q > 2 dim X and any torsion sheaf F on X. We prove this by induction on dim X. If dim X = 0, Γ(X, −) is an exact functor on the category of sheaves on X, our assertion is clear. Suppose that dim X = n ≥ 1 and our assertion holds for those schemes over k of dimensions < n. We may assume that X is reduced. Let η1 , . . . , ηn be those generic points of X so that tr.deg(k(ηi )/k) = n, where tr.deg denotes the transcendental degree. Since n ≥ 1, for each i, we can choose ti ∈ k(ηi ) which is transcendental over k. Choose an irreducible open affine neighborhood Ui of ηi such that ti can be extended to a section in OX (Ui ), which we still denote by ti . The k-algebra homomorphism k[t] → OX (Ui ),

t 7→ ti

defines a k-morphism Ui → A1k . By 1.5.6 (i), we may assume that the morphism is flat by shrinking Ui . Since tr.deg(k(ηi )/k(ti )) = n − 1, the generic fiber of Ui → A1k has dimension ≤ n − 1. If x ∈ Ui lies above a closed point y in A1k , then by [Matsumura (1970)] (13.B) Theorem 19 (2), we have dim(OUi ,x ⊗OA1 ,y k(y)) = dim OUi ,x − dim OA1k ,y ≤ n − 1. k

A1k

So the fibers of Ui → above closed points of A1k have dimensions ≤ n−1. Shrinking Ui again, we may assume that they are disjoint. Let U = ∪i Ui . Then we have a morphism f : U → A1k whose fibers have dimensions ≤ n−1. By the induction hypothesis, we have Rq f! (F |U ) = 0 for any q > 2n − 2. By our construction, we have dim(X − U ) ≤ n − 1. So by the induction hypothesis, we have Hcq (X − U, F |X−U ) = 0

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for any q > 2n − 2. We have a long exact sequence · · · → Hcq (U, F |U ) → Hcq (X, F ) → Hcq (X − U, F |X−U ) → · · · . To prove Hcq (X, F ) = 0 for all q > 2n, it suffices to prove Hcq (U, F |U ) = 0 for any q > 2n. We have a biregular spectral sequence E2pq = Hcp (A1k , Rq f! (F |U )) ⇒ Hcp+q (U, F |U ). Since Rq f! (F |U ) = 0 for any q > 2n − 2, to prove Hcq (U, F |U ) = 0 for any q > 2n, it suffices to prove Hcp (A1k , G ) = 0 for any p > 2 and any torsion sheaf G on A1k . Let j : A1k ,→ P1k be the open immersion. We have Hcp (A1k , G ) ∼ = H p (P1k , j! G ). Our assertion follows from 7.2.7.

Remark 7.4.6. Suppose that f : X → Y is an S-compactifiable morphism. Then the fibers of f have bounded dimensions. By 7.4.5, Rf∗ has finite cohomological dimension if f is proper. So we can define Rf∗ : Dtor (X) → Dtor (Y ) if f is proper, and Rf! : Dtor (X) → Dtor (Y ) in general. Theorem 7.4.7 (Projection formula). Let f : X → Y be an Scompactifiable morphism, and let A be a torsion ring. (i) For any K ∈ ob D− (X, A) and L ∈ ob D− (Y, A), we have a canonical isomorphism ∼ =

∗ L L ⊗L A Rf! K → Rf! (f L ⊗A K). b b (ii) For any K ∈ ob Dtf (X, A), we have Rf! K ∈ ob Dtf (Y, A), and we have a canonical isomorphism ∼ =

∗ L L ⊗L A Rf! K → Rf! (f L ⊗A K)

for any L ∈ ob D(Y, A). (iii) Let A → B be a homomorphism of torsion rings. For any K ∈ ob D− (X, A) and M ∈ ob D− (Y, B), we have a canonical isomorphism ∼ =

∗ L M ⊗L A Rf! K → Rf! (f M ⊗A K)

in D− (Y, B).

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Proof. We prove (i) and leave it for the reader to prove the rest. (Confer 6.5.6 and 6.6.7.) Let j

f¯

X ,→ X → Y

be a compactification of f . For any complexes of sheaves of A-modules M · on X and N · on X, we have a canonical isomorphism j! (j ∗ N · ⊗A M · ) ∼ = N · ⊗A j! M · .

Taking M · to be a complex of flat sheaves of A-modules quasi-isomorphic to K and N · a complex quasi-isomorphic to f¯∗ L, we get an isomorphism ∼ f¯∗ L ⊗L j! K. j! (j ∗ f¯∗ L ⊗L K) = A

A

We have L ∼ ¯ L ⊗L A Rf! K = L ⊗A Rf∗ j! K, ∗ ¯∗ L L ∼ ¯ ∼ ¯ ¯∗ Rf! (f ∗ L ⊗L A K) = Rf∗ j! (j f L ⊗A K) = Rf∗ (f L ⊗A j! K). To prove (i), it suffices to show that the canonical morphism L ⊗L Rf¯∗ j! K → Rf¯∗ (f¯∗ L ⊗L j! K) A

A

is an isomorphism. For any geometric point s → Y , where s is the spectrum of a separably closed field, we need to show that the induced morphism L ¯ ¯ ¯∗ (L ⊗L A Rf∗ j! K)s → (Rf∗ (f L ⊗A j! K))s

on stalks is an isomorphism. Let f¯s : X s = X ×Y s → s ¯ be the base change of f . By the proper base change theorem 7.3.2, we have (L ⊗L Rf¯∗ j! K)s ∼ = Ls ⊗L Rf¯s∗ ((j! K)| ), A

A

Xs

L ∼ ¯ ¯∗ (Rf¯∗ (f¯∗ L ⊗L A j! K))s = Rfs∗ (fs Ls ⊗A (j! K)|X s ). So it suffices to prove that the canonical morphism Ls ⊗L Rf¯s∗ ((j! K)| ) → Rf¯s∗ (f¯∗ Ls ⊗L (j! K)| ) A

s

Xs

A

Xs

is an isomorphism. This follows from 6.5.5 since H i (Ls ) are constant sheaves on s for all i. Corollary 7.4.8. Consider a Cartesian diagram of S-schemes g0

X ×Z Y → X f0 ↓ ↓f g Y → Z. Suppose that g is S-compactifiable and X is quasi-compact and quasiseparated. Let A be a torsion ring. For any K ∈ ob D− (X, A) and L ∈ ob D− (Y, A), we have a canonical isomorphism ∼ =

∗ 0 0∗ L 0∗ K ⊗L A f Rg! L → Rg! (g K ⊗A f L).

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Proof.

By 7.4.4 (i) and 7.4.7 (i), we have ∗ L 0 0∗ ∼ 0 0∗ L 0∗ ∼ K ⊗L A f Rg! L = K ⊗A Rg! f L = Rg! (g K ⊗A f L).

Corollary 7.4.9 (K¨ unneth formula). Let Xi , Yi (i = 1, 2) and Z be Sschemes, let fi : Xi → Yi and Yi → Z be S-compactifiable morphisms, and let pi : X1 ×Z X2 → Xi and qi : Y1 ×Z Y2 → Yi be the projections. For any Ki ∈ ob D− (Xi , A), we have ∗ L ∗ ∗ ∼ q1∗ Rf1! K1 ⊗L A q2 Rf2! K2 = R(f1 × f2 )! (p1 K1 ⊗A p2 K2 ).

Proof. Fix notation by the following commutative diagram of Cartesian squares: f 00

b00

f0

q2

1 1 X1 × Z X2 → Y1 ×Z X2 → X2 00 0 f2 ↓ f2 ↓ ↓ f2 1 X1 ×Z Y2 → Y1 ×Z Y2 → Y2 00 b2 ↓ q1 ↓ ↓ b2

X1

f1

→

b

1 → Z.

Y1

By 7.4.8, we have isomorphisms ∗ 0 0∗ ∗ L 00∗ ∼ q1∗ Rf1! K1 ⊗L A q2 Rf2! K2 = Rf2! (f2 q1 Rf1! K1 ⊗A b1 K2 ) ∼ = Rf 0 Rf 00 (f 00∗ b00∗ K1 ⊗L f 00∗ b00∗ K2 ). 2!

1!

2

2

A

1

Our assertion follows.

1

Let X be a scheme, A a ring, F and G sheaves of A-modules on X, C (F ) and C · (G ) the Godement resolutions of F and G , respectively. By 6.4.9, we have a quasi-isomorphism ·

F ⊗A G → C · (F ) ⊗A C · (G ). Let C · (F ) ⊗A C · (G ) → I · be a quasi-isomorphism so that I · is a complex of injective sheaves of Amodules. Suppose that RΓ(X, −) has finite cohomological dimension. We define the cup product RΓ(X, F ) ⊗L A RΓ(X, G ) → RΓ(X, F ⊗A G )

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to be the composite of the following morphisms · L · ∼ RΓ(X, F ) ⊗L A RΓ(X, G ) = Γ(X, C (F )) ⊗A Γ(X, C (G ))

→ Γ(X, C · (F )) ⊗A Γ(X, C · (G ))

→ Γ(X, C · (F ) ⊗A C · (G )) → Γ(X, I · ) ∼ = RΓ(X, F ⊗A G ). The cup product induces homomorphisms

H i (X, F ) ⊗A H j (X, G ) → H i+j (X, F ⊗A G ). For any s ∈ H i (X, F ) and t ∈ H j (X, G ), denote by s∪t ∈ H i+j (X, F ⊗A G ) the image of s ⊗ t under the above homomorphism. Proposition 7.4.10. Let X be a scheme, let A be a ring, and let F a sheaf of A-modules on X. Denote by τ : H n (X, F ⊗A F ) → H n (X, F ⊗A F ) the homomorphisms induced by τ : F ⊗A F → F ⊗A F ,

s1 ⊗ s2 7→ s2 ⊗ s1

for all n. For any s ∈ H i (X, F ) and t ∈ H j (X, F ), we have s ∪ t = (−1)ij τ (t ∪ s) in H i+j (X, F ⊗A F ). Proof. First note that if K · and L· are two complexes of sheaves of Amodules, then the morphisms σ : K i ⊗A Lj → Lj ⊗A K i ,

x ⊗ y 7→ (−1)ij y ⊗ x

induce a morphism between the complexes associated to the bicomplexes K · ⊗A L· and L· ⊗A K · . Let C · (F ) be the Godement resolution of F and let I · be an injective resolution of C · (F ) ⊗A C · (F ). Consider the following diagram F ⊗A F → C · (F ) ⊗A C · (F ) → I · τ ↓ ↓σ ρ↓ F ⊗A F → C · (F ) ⊗A C · (F ) → I · , where the middle vertical arrow σ is the morphism defined above by taking K · = L· = C · (F ), and ρ : I → I is chosen to make the square on the

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right commute (up to homotopy). One can show that such ρ exists. (Confer the proof of 6.2.8.) So we have a commutative diagram H i+j (Γ(X, C · (F ) ⊗A C · (F ))) → H i+j (Γ(X, I · )) σ ↓ ρ ↓ i+j · · i+j H (Γ(X, C (F ) ⊗A C (F ))) → H (Γ(X, I · )). On the other hand, one can check the following diagram commutes H i (Γ(X, C · (F ))) ⊗A H j (Γ(X, C · (F ))) → H i+j (Γ(X, C · (F ) ⊗A C · (F ))) σ ↓ ↓σ j · i · i+j · H (Γ(X, C (F ))) ⊗A H (Γ(X, C (F ))) → H (Γ(X, C (F ) ⊗A C · (F ))), where the first vertical arrow is the homomorphism s ⊗ t 7→ (−1)ij t ⊗ s. So we have a commutative diagram H i (Γ(X, C · (F ))) ⊗A H j (Γ(X, C · (F ))) → H i+j (Γ(X, I · )) σ ↓ ρ↓ H j (Γ(X, C · (F ))) ⊗A H i (Γ(X, C · (F ))) → H i+j (Γ(X, I · )). This diagram can be identified with H i (X, F ) ⊗A H j (X, F ) → H i+j (X, F ⊗A F ) σ ↓ τ ↓ H j (X, F ) ⊗A H i (X, F ) → H i+j (X, F ⊗A F ). Our assertion follows.

Proposition 7.4.11. Let X and Y be proper schemes over a separably closed field k, A a torsion ring, F a flat sheaf of A-modules on X, and G a sheaf of A-modules on Y . Denote by p : X ×k Y → X and q : X ×k Y → Y the projections. Then the homomorphism H 2 dim X (X, F ) ⊗A H 2 dim Y (Y, G ) → H 2(dim X+dim Y ) (X ×k Y, p∗ F ⊗A q ∗ G ), s ⊗ t 7→ p∗ s ∪ q ∗ t

is an isomorphism. Suppose that either H i (X, F ) are flat A-modules for all i, or H i (Y, G ) are flat A-modules for all i. Then the homomorphisms M (H i (X, F ) ⊗A H j (Y, G )) → H v (X ×k Y, p∗ F ⊗A q ∗ G ), i+j=v

s ⊗ t 7→ p∗ s ∪ q ∗ t

are isomorphisms for all v.

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Proof.

By the K¨ unneth formula 7.4.9, we have ∗ L ∗ ∼ RΓ(X, F ) ⊗L A RΓ(Y, G ) = RΓ(X ×k Y, p F ⊗A q G ).

Since F is flat, we have ∗ ∗ ∼ ∗ p∗ F ⊗ L A q G = p F ⊗A q G .

By 6.4.12, we have a biregular spectral sequence M E2uv = Tor−u (H i (X, F ), H j (Y, G )) ⇒ H u+v (RΓ(X, F )⊗L A RΓ(Y, G )). i+j=v

By 7.4.5, we have H i (X, F ) = 0 for any i > 2 dim X, and H j (Y, G ) = 0 for any j > 2 dim Y . It follows that E2uv = 0 for u > 0 or v > 2(dim X +dim Y ). So we have 2(dim X+dim Y ) H 2 dim X (X, F ) ⊗A H 2 dim Y (Y, G ) ∼ (RΓ(X, F ) ⊗L =H A RΓ(Y, G )).

Hence H 2 dim X (X, F )⊗A H 2 dim Y (Y, G ) ∼ = H 2(dim X+dim Y ) (X ×k Y, p∗ F ⊗A q ∗ G ). If either H i (X, F ) are flat A-modules for all i, or H i (Y, G ) are flat Amodules for all i, then the above spectral sequence degenerates, and we have M (H i (X, F ) ⊗A H j (Y, G )) ∼ = H v (RΓ(X, F ) ⊗L A RΓ(Y, G )) i+j=v

∼ = H v (X ×k Y, p∗ F ⊗A q ∗ G ).

To prove our assertion, it suffices to show that these isomorphisms are induced by cup products. By 6.4.10, we have quasi-isomorphisms p∗ F ⊗A q ∗ G → p∗ C · (F ) ⊗A q ∗ C · (G ) → C · (p∗ F ) ⊗A C · (q ∗ G ). Here we use the functorial Godement resolutions constructed in the end of 5.6. Let f : X → Spec k and g : Y → Spec k be the structure morphisms. The isomorphism ∗ L ∗ ∼ RΓ(X, F ) ⊗L A RΓ(Y, G ) = RΓ(X ×k Y, p F ⊗A q G )

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defined by the K¨ unneth formula is the composite of the following morphisms: · L · ∼ RΓ(X, F ) ⊗L A RΓ(Y, G ) = Γ(X, C (F )) ⊗A Γ(Y, C (G ))

→ Γ(X, C · (F )) ⊗A Γ(Y, C · (G ))

→ Γ(Y, g ∗ f∗ C · (F )) ⊗A Γ(Y, C · (G )) → Γ(Y, g ∗ f∗ C · (F ) ⊗A C · (G )) → Γ(Y, q∗ p∗ C · (F ) ⊗A C · (G ))

→ Γ(Y, q∗ (p∗ C · (F ) ⊗A q ∗ C · (G )))

= Γ(X ×k Y, p∗ C · (F ) ⊗A q ∗ C · (G ))

→ RΓ(X ×k Y, p∗ C · (F ) ⊗A q ∗ C · (G )) ∗ ∼ = RΓ(X ×k Y, p∗ F ⊗L A q G ). The composite of the above arrows from the third to the seventh coincide with the composite Γ(X, C · (F )) ⊗A Γ(Y, C · (G ))

→ Γ(X ×k Y, p∗ C · (F )) ⊗A Γ(X ×k Y, q ∗ C · (G )) → Γ(X ×k Y, p∗ C · (F ) ⊗A q ∗ C · (G )).

So the isomorphism defined by the K¨ unneth formula is the composite of the following morphisms: · L · ∼ RΓ(X, F ) ⊗L A RΓ(Y, G ) = Γ(X, C (F )) ⊗A Γ(Y, C (G ))

→ Γ(X, C · (F )) ⊗A Γ(Y, C · (G ))

→ Γ(X ×k Y, p∗ C · (F )) ⊗A Γ(X ×k Y, q ∗ C · (G )) → Γ(X ×k Y, p∗ C · (F ) ⊗A q ∗ C · (G ))

→ RΓ(X ×k Y, p∗ C · (F ) ⊗A q ∗ C · (G )) ∗ ∼ = RΓ(X ×k Y, p∗ F ⊗L A q G ).

The morphism defined via the cup product p∗ ⊗q∗

∗ L ∗ RΓ(X, F ) ⊗L A RΓ(Y, G ) → RΓ(X ×k Y, p F ) ⊗A RΓ(X ×k Y, p G ) ∪

∗ → RΓ(X ×k Y, p∗ F ⊗L A q G)

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is the composite of the following morphisms: · L · ∼ RΓ(X, F ) ⊗L A RΓ(Y, G ) = Γ(X, C (F )) ⊗A Γ(Y, C (G ))

→ Γ(X, C · (F )) ⊗A Γ(Y, C · (G ))

→ Γ(X ×k Y, p∗ C · (F )) ⊗A Γ(X ×k Y, q ∗ C · (G )) → Γ(X ×k Y, C · (p∗ F )) ⊗A Γ(X ×k Y, C · (q ∗ G )) → Γ(X ×k Y, C · (p∗ F ) ⊗A C · (q ∗ G ))

→ RΓ(X ×k Y, C · (p∗ F ) ⊗A C · (q ∗ G )) ∗ ∼ = RΓ(X ×k Y, p∗ F ⊗L A q G ).

The coincidence of the isomorphism defined by the K¨ unneth formula and the morphism defined via the cup product follows from the commutativity of the following diagram: Γ(X ×k Y, p∗ C · (F )) ⊗A Γ(X ×k Y, q ∗ C · (G )) → Γ(X ×k Y, p∗ C · (F ) ⊗A q ∗ C · (G )) ↓ ↓ Γ(X ×k Y, C · (p∗ F )) ⊗A Γ(X ×k Y, C · (q ∗ G )) →Γ(X ×k Y, C · (p∗ F ) ⊗A C · (q ∗ G )).

7.5

Cohomological Dimension of Rf∗

([SGA 4] XIV 2–4.) Theorem 7.5.1. Let X and Y be schemes of finite type over a field k, f : X → Y an affine k-morphism, F a torsion sheaf on X, and n an integer. Suppose for any point a ∈ X with the property dim {a} > n, we have Fa¯ = 0. Then for any point b ∈ Y with the property dim {b} > n − q, we have (Rq f∗ F )¯b = 0. Taking Y to be the spectrum of a separably closed field, we get the following: Corollary 7.5.2. Let X be an affine scheme of finite type over a separably closed field k, F a torsion sheaf on X, and n an integer. Suppose for any point a ∈ X with the property dim {a} > n, we have Fa¯ = 0. Then H q (X, F ) = 0 for any q > n. In particular, we have H q (X, F ) = 0 for any q > dim X and any torsion sheaf F on X. To prove 7.5.1, we need the following lemma. Lemma 7.5.3. Let k be a field, X → Spec k[t1 , . . . , td ] a morphism of finite type, K an algebraic extension of k(t1 , . . . , td ), and XK = X ⊗k[t1 ,...,kd ] K.

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For any point a0 ∈ XK , let a be its image in X. We have dim {a} = dim {a0 } + d, where {a} (resp. {a0 }) denotes the closure of {a} (resp. {a0 }) in X (resp. XK ). Moreover, we have dim XK ≤ dim X − d. Spec K XK = X ⊗k[t1 ,...,td ] K → ↓ ↓ X → Spec k[t1 , . . . , td ] → Spec k. Proof.

We have dim {a0 } = tr.deg(k(a0 )/K),

dim {a} = tr.deg(k(a)/k),

where tr.deg denotes the transcendental degree. Note that k(a0 ) is algebraic over k(a). So we have tr.deg(k(a)/k) = tr.deg(k(a0 )/k) = tr.deg(k(a0 )/K) + tr.deg(K/k) = tr.deg(k(a0 )/K) + d, and hence dim {a} = dim {a0 } + d. Let ηi0 be all the generic points of XK , and let ηi be their images in X. We have dim XK = max dim {ηi0 } = max dim {ηi } − d ≤ dim X − d. i

i

Proof of 7.5.1. Making the base change from k to its algebraic closure, we may assume that k is algebraically closed. Replacing X and Y by Xred and Yred , we may assume that X and Y are reduced. The problem is local with respect to Y . We may assume that Y is affine. Then X is also affine. We have F = limλ Fλ , where Fλ are constructible subsheaves of F . Fλ −→ satisfy the same condition as F , and Rq f∗ F = lim Rq f∗ Fλ . −→ λ

So we may assume that F is constructible. There is a decomposition X = ∪i Xi such that Xi are finitely many irreducible locally closed subschemes of X and F |Xi are locally constant. If Fx¯ 6= 0 for one x ∈ Xi , then Fx¯ 6= 0 for all x ∈ Xi . We then have dim Xi ≤ n by our assumption. This shows that dim(supp F ) ≤ n, where supp F is the support of F .

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We use induction on n. Suppose n = 0. Then supp F consists of finitely many closed points in X. Put the reduced closed subscheme structure on supp F , and let i : supp F → X be the closed immersion. We have R q f∗ F ∼ = Rq (f i)∗ i∗ F .

But f i is a finite morphism. So Rq (f i)∗ i∗ F = 0 for all q ≥ 0 and (f i)∗ i∗ F is supported on finitely many closed points of Y . In particular, we have (Rq f∗ F )¯b = 0 for any b ∈ Y satisfying dim {b} > −q. Suppose n = 1. Replacing X by supp F , we may assume dim X = 1. We have (Rq f∗ F )¯b ∼ H q (X ×Y U, F |X×Y U ), = lim −→ U

where U → Y goes over the family of affine etale neighborhood of ¯b. Since f is affine, each X ×Y U is affine. We have dim (X ×Y U ) ≤ 1. By 7.2.13, we have H q (X ×Y U, F |X×Y U ) = 0

for any q ≥ 2. So (Rq f∗ F )¯b = 0 for any q > 1 − dim {b} if dim {b} = 0. We have supp(Rq f∗ F ) ⊂ f (X). Note that dim f (X) ≤ 1. To prove this, let Xi be an arbitrary irreducible component of X. Then f (Xi ) is also irreducible. The generic point ξi of Xi is mapped to the generic point of f (Xi ). It follows that dim f (Xi ) = tr.deg(k(f (ξi ))/k) ≤ tr.deg(k(ξi )/k) = dim Xi ≤ 1.

So we have dim f (X) ≤ 1. If dim f (X) = 0, we have (Rq f∗ F )¯b = 0 for all q and those b with dim {b} > 0. Suppose dim f (X) = 1. Then (Rq f∗ F )¯b = 0 for all q if dim {b} > 1. It remains to show that for any b ∈ f (X) with dim {b} = 1, we have (R1 f∗ F )¯b = 0. Put the reduced closed subscheme structure on f (X). Let g : X → f (X) be the morphism induced by f and let i : f (X) → Y be the closed immersion. We have R 1 f∗ F = i∗ R 1 g∗ F .

We claim that there exists an open subset U of f (X) such that f (X) − U consists of finitely many closed point and that gU : g −1 (U ) → U is finite. Then (R1 g∗ F )|U = 0. For any b ∈ f (X) with dim {b} = 1, we have b ∈ U

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and hence (R1 f∗ F )¯b = 0. Let η1 , . . . , ηm be all the generic points of f (X) with dim{ηj } = 1. By 1.10.10 (iv), to prove our claim, it suffices to show that g −1 (ηj ) → ηj is finite for each j. Since g maps closed points of X to closed points of f (X) and none of the ηj is a closed point, points in g −1 (ηj ) are generic points of irreducible components of X of dimension 1. The residue fields of X at such kind of generic points and the residue field of Y at ηj have transcendental degree 1 over k. It follows that the residue field of X at any point in g −1 (ηj ) is algebraic and hence finite over the residue field of Y at ηj . So g −1 (ηj ) → ηj is finite. This proves the theorem for the case n = 1. Suppose n ≥ 2, and suppose that the theorem holds if n is replaced by any integer strictly lesser than n. Since X and Y are affine, f : X → Y can be factorized as a composite i

π

X → Am Y →Y such that i is a closed immersion and π is the projection. We have R q f∗ F ∼ = Rq π∗ (i∗ F ) and i∗ F satisfies the same condition as F . We are thus reduced to the case where f is the projection π : AnY → Y . Note that if f = gh and if the theorem holds for g and h, then it holds for f . Indeed, we have a biregular spectral sequence E2pq = Rp g∗ Rq h∗ F ⇒ Rp+q f∗ F .

To prove (Rq f∗ F )¯b = 0 for any b with dim {b} > n − q, it suffices to prove (Rp g∗ Rq h∗ F )¯b = 0

for any b with dim {b} > n − (p + q). Since the theorem holds for h, we have (Rq h∗ F )c¯ = 0 for any c with dim {c} > n − q. Since the theorem holds for g, we have (Rp g∗ Rq h∗ F )¯b = 0 for any b with dim {b} > n − q − p. This proves our assertion. The projection π : Am Y → Y is the composite of projections m−1 Am → · · · → A1Y → Y. Y → AY

So it suffices to consider the case where f is the projection A1Y → Y . Let Y = Spec B for some finitely generated k-algebra B, let b ∈ Y , and let q be the prime ideal of B corresponding to b. We need to show that (Rq f∗ F )¯b = 0 for any q > n − dim {b}. First consider the case where dim {b} > 0. We have dim {b} = tr.degk (Bq /qBq ),

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where tr.degk denotes the transcendental degree over k. Replacing Y by an affine open neighborhood of b, we may assume that there exist t1 , . . . , td ∈ B such that their images in Bq /qBq form a separating transcendental basis for Bq /qBq over k, where d = dim {b} > 0. (We have reduced to the case where k is algebraically closed. Hence a separating transcendental basis exists.) We have q ∩ k[t1 , . . . , td ] = 0. Let K be a separable closure of Bq /qBq and let Y 0 = Spec (B ⊗k[t1 ,...,td ] K). There exists a point b0 ∈ Y 0 lying above b. Let X 0 = A1Y 0 , let f 0 : X 0 → Y 0 be the projection, and let F 0 be the inverse image of F in X 0 . f0

X 0 = A1Y 0 → Y 0 = Spec(B ⊗k[t1 ,...,td ] K) → ↓ ↓ f

X = A1Y →

Y = Spec B

Spec K ↓

→ Spec k[t1 , . . . , td ] → Spec k.

Since K is separable over k(t1 , . . . , td ), the strict localization of Y 0 at ¯b0 is isomorphic to the strict localization of Y at ¯b. So by 5.9.5, we have (Rq f∗ F )¯b ∼ = (Rq f∗0 F 0 )¯b0 . For any a0 ∈ X 0 , let a be its image in X. We have dim {a} = dim {a0 } + d by 7.5.3. Since Fa¯ = 0 for any a ∈ X with dim {a} > n, we have Fa¯0 0 = 0 for any a0 ∈ X 0 with dim {a0 } > n − d. But n − d ≤ n − 1. So we can apply the induction hypothesis to F 0 and to the morphism f 0 : X 0 → Y 0 , and we get (Rq f∗0 F 0 )¯b0 = 0 if dim {b0 } > n − d − q. So we have (Rq f∗ F )¯b = 0 if dim {b} > n − q. Next we consider the case where dim {b} = 0. We need to show that (Rq f∗ F )¯b = 0 for any q > n. Let j : X = A1Y ,→ P1Y be the open immersion and let f¯ : P1Y → Y be the projection. We have a biregular spectral sequence E2pq = Rp f¯∗ Rq j∗ F ⇒ Rp+q f∗ F . It suffices to show that (Rp f¯∗ Rq j∗ F )¯b = 0 for any p + q > n. Recall that n ≥ 2. If q = 0, we have Rp f¯∗ Rq j∗ F = Rp f¯∗ j∗ F = 0

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for any p > 2 by 7.4.5. For any q > 0, we have (Rq j∗ F )|A1Y = 0. Since f¯|P1Y −A1Y is an isomorphism, we have

Rp f¯∗ Rq j∗ F = 0

for any p, q > 0, and f¯∗ Rq j∗ F can be identified with (Rq j∗ F )|P1Y −A1Y for any q > 0. To prove our assertion, it suffices to prove (Rq j∗ F )a¯ = 0 for any q > n and any point a in P1Y − A1Y . Put the reduced closed subscheme structure on the closure supp F of supp F in P1Y , and let j 0 : supp F ∩ A1Y ,→ supp F be the base change of j. If a 6∈ supp F , we have (Rq j∗ F )a¯ = 0 for all q. If a ∈ supp F , we have (Rq j∗ F )a¯ ∼ = (Rq j∗0 (F |supp F ∩A1 ))a¯ . Y

Let R be the strict henselization of Osupp F ,a . We have (Rq j∗ F )a¯ ∼ = H q (A1Y ×P1Y Spec R, F ) ∼ = H q (A1k ×P1k Spec R, F ). Here we denote also by F the inverse images of F on A1Y ×P1Y Spec R and on A1k ×P1k Spec R. We need to prove that H q (A1k ×P1k Spec R, F ) = 0 for any q > n. Choose an inverse system {Uα } of a sufficiently small affine etale neighborhood of a ¯ in supp F such that R = limα Γ(Uα , OUα ). We have −→ q 1 ∼ H (Ak ×P1k Spec R, F ) = lim H q (A1k ×P1k Uα , F ). −→ α e1 be the strict localization The point a is above the point ∞ of P1k . Let P k∞ 1 e1 . We have a commutative of Pk at ∞, and let η˜∞ be the generic point of P k∞ diagram e1 Spec R → P k∞ ↓ ↓ Uα → P1k .

Making the base change A1k → P1k , we obtain a commutative diagram A1k ×P1k Spec R → η˜∞ ↓ ↓ A1k ×P1k Uα → A1k . The homomorphism H q (A1k ×P1k Uα , F ) → H q (A1k ×P1k Spec R, F )

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factors through H q (˜ η∞ ×A1k (A1k ×P1k Uα ), F ) → H q (A1k ×P1k Spec R, F )

To prove our assertion, it suffices to show H q (˜ η∞ ×A1k (A1k ×P1k Uα ), F ) = 0 for all q > n. Since dim supp F ≤ n, we have dim Uα ≤ n, and hence dim(A1k ×P1k Uα ) ≤ n. By 7.5.3, we have dim(˜ η∞ ×A1k (A1k ×P1k Uα )) ≤ n − 1.

Let g : η˜∞ ×A1k (A1k ×P1k Uα ) → η˜∞

be the projection. Note that g is an affine morphism. Indeed, A1k → P1k is an affine morphism. So the projection A1k ×P1k Uα → Uα is an affine morphism. Since Uα is affine, A1k ×P1k Uα is also affine. So the morphism A1k ×P1k Uα → A1k is affine, and hence g is affine. By the induction hypothesis, we have Rq g∗ (F |η˜∞ ×A1 (A1k ×P1 Uα ) ) = 0 k

k

for all q > n − 1. Since the residue field of η˜∞ has transcendental degree 1 over the algebraically closed field k, by 4.5.11, we have H p (˜ η∞ , Rq g∗ (F |η˜∞ ×A1 (A1k ×P1 Uα ) )) = 0 k

k

for all p > 1. We have a biregular spectral sequence E2pq = H p (˜ η∞ , Rq g∗ (F |η˜∞ ×A1 (A1k ×P1 Uα ) )) ⇒ H p+q (˜ η∞ ×A1k (A1k ×P1k Uα ), F ). k

k

It follows that H q (˜ η∞ ×A1k (A1k ×P1k Uα ), F ) = 0 for all q > n. This proves our assertion. Lemma 7.5.4. Let k be an algebraically closed field, X an integral scheme of finite type over k, n = dim X, η the generic point of X, F a torsion sheaf on η, and j : η → X the canonical morphism. Then we have (Rq j∗ F )a¯ = 0 for any a ∈ X with dim {a} > n − q. Proof. Replacing X by an open affine neighborhood of a does not change (Rq j∗ F )a¯ . So we may assume that X = Spec A for some finitely generated k-algebra A. Let p be the prime ideal of A corresponding to a, and let d = dim {a}. We have tr.degk (Ap /pAp ) = d. Shrinking X, we may assume that there exist t1 , . . . , td ∈ A such that their images in Ap /pAp form a separating transcendental basis for Ap /pAq over k. We have p ∩ k[t1 , . . . , td ] = 0.

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Let K be a separable closure of Ap /pAp and let X 0 = Spec (A⊗k[t1 ,...,td ] K). There exists a point a0 ∈ X 0 lying above a. Since K is separable over e 0 0 of X 0 at a k(t1 , . . . , td ), the strict localization X ¯0 is isomorphic to the a ¯ ea¯ of X at a strict localization X ¯. By 5.9.5, we have q q ∼ ea¯ , F ) ∼ ea¯0 0 , F ). (R j∗ F )a¯ = H (η ×X X = H q (η ×X X

e 0 0 at its generic points. We have Let K1 , . . . , Km be the residue fields of X a ¯ a a ea¯0 0 ∼ η ×X X Spec K · ·· Spec Km , = 1

and

tr.deg(Ki /K) = tr.deg(Ki /k) − tr.deg(K/k)

= tr.deg(k(η)/k) − tr.deg(K/k)

By 4.5.10, we have

= n − d.

H q (Spec Ki , F ) = 0 for all q > n − d. So we have (Rq j∗ F )a¯ = 0 for all q > n − d. Theorem 7.5.5. Let k be a separably closed field, X a type over k, n an integer, F a torsion sheaf on X such any a ∈ X with dim {a} > n. Then H q (X, F ) = 0 for particular, we have H q (X, F ) = 0 for any q > 2 dim X sheaf F on X.

scheme of finite that Fa¯ = 0 for any q > 2n. In and any torsion

Proof. We may assume that k is algebraically closed and F is constructible. We use induction on n. If n = 0, then supp F consists of finitely many closed point of X, and we have H q (X, F ) = 0 for all q > 0. Suppose that the theorem holds if n is replaced by any integer strictly less than n. Let ηv be those points in supp F with dim {ηv } = n, let jv : Spec k(ηv ) → X be the canonical morphisms, and let K (resp. C ) be the kernel (resp. cokernel) of the canonical morphism Y F → jv∗ jv∗ F . v

We have

Ka¯ = Ca¯ = 0 for any a in X with dim {a} > n − 1. Applying the induction hypothesis to K and C , we see that H q (X, K ) and H q (X, C ) vanish for any q > 2(n−1). To prove H q (X, F ) = 0 for any q > 2n, it suffices to show that H q (X, jv∗ jv∗ F ) = 0

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for any q > 2n. Replacing X by the reduced closed subscheme {ηv }, we are reduced to prove the following statement: Let X be an integral scheme of finite type over an algebraically closed field k of dimension n, let η be the generic point of X, and let j : η → X be the canonical morphism. Then for any torsion sheaf F on η, we have H q (X, j∗ F ) = 0 for any q > 2n. We have a biregular spectral sequence E2pq = H p (X, Rq j∗ F ) ⇒ H p+q (η, F ).

By 7.5.4, we have (Rq j∗ F )a¯ = 0 if dim {a} > n − q. For each q > 0, we can apply the induction hypothesis to Rq j∗ F , and we get E2pq = H p (X, Rq j∗ F ) = 0 if q > 0 and p > 2(n − q). This implies that E p0 ∼ = H p (η, F ) 2

for any p > 2n. We have tr.deg(k(η)/k) = n. By 4.5.10, we have H p (η, F ) = 0 for any p > n. So we have H p (X, j∗ F ) = E2p0 = 0 for any p > 2n.

Corollary 7.5.6. Let k be a field, X and Y schemes of finite type over k, and f : X → Y a k-morphism. Then Rq f∗ F = 0 for any q > 2 dim X and any torsion sheaf F on X. In particular, Rf∗ has finite cohomological dimension on the category of torsion sheaves. Proof. We may assume that k is algebraically closed. Rq f∗ F is the sheaf associated to the presheaf V 7→ H q (V ×Y X, F )

for any etale X-scheme V . By 7.5.5, we have for any q > 2 dim X.

H q (V ×Y X, F ) = 0

Corollary 7.5.7. Let f : X → S be a morphism of finite type, x ∈ X, Sef (¯x) the strict localization of S at f (¯ x), j : t → Sef (¯x) a geometric point, and e ˜j : X ×S t → X ×S Sf¯(x) the morphism induced by j by base change. Then Rq ˜j∗ F = 0 for any q > dim(X ×S t) and any torsion sheaf F on X ×S t. In particular, R˜j∗ has finite cohomological dimension on the category of torsion sheaves.

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Proof.

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For each q, Rq ˜j∗ F is the sheaf associated to the presheaf V 7→ H q (V ×Sef (¯x) t, F )

for any etale X ×S Sef (¯x) -scheme V . We have

dim(V ×Sef (¯x) t) ≤ dim(X ×S t).

If V is affine, V ×Sef (¯x) t is also affine, and we have H q (V ×Sef (¯x) t, F ) = 0 for any q > dim(X ×S t) by 7.5.1. So we have Rq ˜j∗ F = 0 for any q > dim(X ×S t). 7.6

Local Acyclicity

([SGA 4 21 ] Arcata V 1, Th. finitude 2.12–2.16, Appendice 2.9, 2.10.) Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S → S.

Let A be a ring. Suppose that one of the following conditions holds: (a) Rg∗ and Rg∗0 have finite cohomological dimension, K ∈ ob D− (X, A) and L ∈ ob D− (S 0 , A). b (b) K ∈ ob Dtf (X, A) has finite Tor-dimension and L ∈ ob D+ (S 0 , A). b (c) Rg∗ and Rg∗0 have finite cohomological dimension, K ∈ ob Dtf (X, A) 0 has finite Tor-dimension and L ∈ ob D(S , A). Then we define a canonical morphism ∗ 0 0∗ L 0∗ K ⊗L A f Rg∗ L → Rg∗ (g K ⊗A f L)

as the composite of canonical morphisms ∗ L 0 0∗ 0 0∗ L 0∗ K ⊗L A f Rg∗ L → K ⊗A Rg∗ f L → Rg∗ (g K ⊗A f L).

By 7.4.8, this canonical morphism is an isomorphism if g is proper. In this section, we give conditions on f under which this canonical morphism is an isomorphism. Let X be a scheme and let t → X be a geometric point in X, where t is the spectrum of a separably closed field. We often denote by k(t) the separably closed field such that t = Spec k(t). Let x be the image of t in

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X. We say that t is an algebraic geometric point if k(t) is algebraic over the residue field k(x). ex¯ Let f : X → S be a morphism. For any x ∈ X (resp. s ∈ S), let X e (resp. Ss¯) be the strict localization of X (resp. S) at x ¯ (resp. s¯). Let K ∈ ob D+ (X). We say that f is locally acyclic at x ∈ X relative to K if for any algebraic geometric point t of Sef (¯x) , the canonical morphism ex¯ , K) → RΓ(X ex¯ × e Kx¯ = RΓ(X t, K) Sf (¯ x)

is an isomorphism. By 5.7.2 (ii), this is equivalent to saying that the above canonical morphism is an isomorphism if t is of the form y¯ for any y ∈ Sef (¯x) . We say f is locally acyclic relative to K if it is locally acyclic relative to K at every point of X. Let s ∈ S and let t be an algebraic geometric point of Ses¯. Fix notation by the following diagram: ˜ j

˜i

Xt = X ×S t → X ×S Ses¯ ← Xs¯ = X ×S s¯ ft ↓ f˜ ↓ ↓ fs¯ j i t→ Ses¯ ← s¯.

e be the where all vertical arrows are induced by f by base change. Let K −1 e inverse image of K on X ×S Ss¯. For any x ∈ f (s), the strict localization of X ×S Ses¯ at x ¯ is isomorphic to the strict localization of X at x ¯, and we have e x¯ ∼ (˜i∗ K) = Kx¯ , ∗ ˜ ˜∗ e e x¯ ∼ ex¯ × e ˜ t, K). (i Rj∗ j K)x¯ ∼ = RΓ(X = (R˜j∗ ˜j ∗ K) Sf (¯ x)

So f is locally acyclic relative to K if and only if for any s ∈ S and any algebraic geometric point t of Ses¯, the canonical morphism e e → ˜i∗ R˜j∗ ˜j ∗ K ˜i∗ K

is an isomorphism. We say that f is universally locally acyclic relative to K if for any morphism S 0 → S, the base change X ×S S 0 → S 0 of f is locally acyclic relative to the inverse image K|X×S S 0 of K. By 5.9.3 or 5.9.6, f is universally locally acyclic relative to K if and only if for any morphism S 0 → S locally of finite type, the base change X ×S S 0 → S 0 of f is locally acyclic relative to the inverse image K|X×S S 0 of K. b Let A be a ring, and let K ∈ ob Dtf (X, A). We say that f is strongly locally acyclic at x relative to K if for any algebraic geometric point t of Sef (¯x) and any A-module M , the canonical morphism L L e e Kx¯ ⊗L ef (x) t, K ⊗A M ) A M = RΓ(Xx , K) ⊗A M → RΓ(Xx ×S

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is an isomorphism. We say that f is strongly locally acyclic relative to K if it is strongly locally acyclic relative to K at every point of X. For any s ∈ S and any algebraic geometric point t of Ses¯, define a canonical morphism e ⊗L f˜∗ Rj∗ M ) → ˜i∗ R˜j∗ (˜j ∗ K e ⊗L f ∗ M ) ˜i∗ (K A A t

to be the composite of canonical morphisms

˜∗ e ⊗L ˜∗ ˜ ˜∗ e L ∗ ˜i∗ (K ˜∗ e L ˜ ∗ A f Rj∗ M ) → i (K ⊗A Rj∗ ft M ) → i Rj∗ (j K ⊗A ft M ).

Lemma 7.6.1. In the above notation, f is strongly locally acyclic relative to K if and only if for any s ∈ S and any algebraic geometric point t of Ses¯, the canonical morphism e ⊗L f ∗ M ) e ⊗L f˜∗ Rj∗ M ) → ˜i∗ R˜j∗ (˜j ∗ K ˜i∗ (K A t A

is an isomorphism.

Proof. In the following, we denote also by M the constant sheaf associated to M on any scheme. Note that the canonical morphism i∗ M → i∗ Rj∗ j ∗ M is an isomorphism. Indeed, for any q, Rq j∗ j ∗ M is the sheaf associated to the presheaf V 7→ H q (V ×Ses¯ t, M ) for any etale Ses¯-scheme V . Since t is the spectrum of a separably closed field, V ×Ses¯ t is a disjoint union of copies of t and hence H q (V ×Ses¯ t, M ) = 0

for any q ≥ 1. So

Rq j∗ j ∗ M = 0

for any q ≥ 1. Moreover, we have

(i∗ j∗ j ∗ M )s¯ ∼ = Γ(Ses¯, j∗ j ∗ M ) ∼ = M.

So we have

i∗ M ∼ = i∗ Rj∗ j ∗ M. This implies that the canonical morphism ∗ ˜∗ e ⊗L ˜i∗ (K ˜∗ e L ˜∗ A f M ) → i (K ⊗A f Rj∗ j M )

is an isomorphism.

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We have a commutative diagram e ⊗L f˜∗ M ) ˜i∗ (R˜j∗ ˜j ∗ K A

% & ∗ e L ∗ ˜∗ e ⊗L f˜∗ M ) e ⊗L ˜j ∗ f˜∗ M ) ˜i∗ (K ˜ ˜ ˜ → i (K ⊗A Rj∗ j f M ) → ˜i∗ R˜j∗ (˜j ∗ K A A ∼ ∼ ∼ = ↓ = ↓ =↓ e ⊗L f˜∗ Rj∗ j ∗ M ) → ˜i∗ (K e ⊗L R˜j∗ ft∗ j ∗ M ) → ˜i∗ R˜j∗ (˜j ∗ K e ⊗L ft∗ j ∗ M ), ˜i∗ (K A A A

where all vertical arrows are isomorphisms. By definition, f is strongly locally acyclic relative to K if and only if the composite e ⊗L f˜∗ M ) → ˜i∗ (R˜j∗ ˜j ∗ K e ⊗L f˜∗ M ) → ˜i∗ R˜j∗ (˜j ∗ K e ⊗L ˜j ∗ f˜∗ M ) ˜i∗ (K A A A

on the upper part of the above diagram defines an isomorphism ˜∗ e ⊗L ˜i∗ (K ˜∗ ˜ ˜∗ e L ˜∗ ˜∗ A f M ) → i Rj∗ (j K ⊗A j f M ).

By the commutativity of the above diagram, this condition is equivalent to saying that the composite e ⊗L f˜∗ Rj∗ j ∗ M ) → ˜i∗ (K e ⊗L R˜j∗ f ∗ j ∗ M ) → ˜i∗ R˜j∗ (˜j ∗ K e ⊗L f ∗ j ∗ M ) ˜i∗ (K t A t A A

on the lower part of the above diagram defines an isomorphism ∗ ˜∗ e ⊗L ˜i∗ (K ˜∗ ˜ ˜∗ e L ∗ ∗ A f Rj∗ j M ) → i Rj∗ (j K ⊗A ft j M ).

This proves our assertion.

If R˜j∗ has finite cohomological dimension for any s ∈ S and any algebraic geometric point j : t → Ses¯, then in the above discussion, we do not need to assume that K has finite Tor-dimension and can talk about the strong local acyclicity of f relative to K for any K ∈ D− (X, A). Many results in this section also hold under this assumption. We say that f is universally strongly locally acyclic relative to K if for any morphism S 0 → S, the base change X ×S S 0 → S 0 of f is strongly locally acyclic relative to the inverse image K|X×S S 0 of K. By 5.9.3 or 5.9.6, f is universally strongly locally acyclic relative to K if and only if for any morphism S 0 → S locally of finite type, the base change X ×S S 0 → S 0 of f is strongly locally acyclic relative to the inverse image K|X×S S 0 of K. Proposition 7.6.2. Let f : X → S be a morphism, A a ring and K ∈ ob D− (X, A). Suppose for any x ∈ X and any algebraic geometric point j : t → Sef (¯x) , that the functor R˜j∗ has finite cohomological dimension, where ˜j : X ×S t → X ×S Sef¯(x) is the morphism induced by j by base change. If f is locally acyclic relative to K, then f is strongly locally acyclic relative to K.

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Note that if f is of finite type and A is a torsion ring, then R˜j∗ has finite cohomological dimension by 7.5.7. Proof. Keep the notation in the proof of 7.6.1. If f is locally acyclic relative to K, we have e ∼ e ˜i∗ K = ˜i∗ R˜j∗ ˜j ∗ K. So for any A-module M , we have e ⊗L f˜∗ M ) ∼ e ⊗L f˜∗ M ). ˜i∗ (K = ˜i∗ (R˜j∗ ˜j ∗ K A A By 6.5.5, we have It follows that

˜∗ ∼ ˜ ˜∗ e L ˜∗ ˜∗ e ⊗L R˜j∗ ˜j ∗ K A f M = Rj∗ (j K ⊗A j f M ). ˜∗ ∼ ˜∗ ˜ ˜∗ e L ˜∗ ˜∗ e ⊗L ˜i∗ (K A f M ) = i Rj∗ (j K ⊗A j f M ).

So f is strongly locally acyclic relative to K.

Proposition 7.6.3. Let A be a ring, let f : X → S be a morphism, and let b K ∈ ob Dtf (X, A). Suppose that f is locally acyclic (resp. strongly locally acyclic) relative to K. Then for any quasi-finite morphism g : S 0 → S, the base change f 0 : X 0 = X ×S S 0 → S 0 of f is locally acyclic (resp. strongly locally acyclic) relative to the inverse image K 0 = K|X 0 of K. Conversely, if f 0 is locally acyclic (resp. strongly locally acyclic) relative to K 0 and g is surjective, then f is locally acyclic (resp. strongly locally acyclic) relative to K. Proof. We prove the statements for local acyclicity. The problem is local with respect to S 0 . So we may assume that g is separated and S is quasicompact and quasi-separated. By the Zariski Main Theorem in [EGA] IV 18.12.13, g is the composite of an open immersion and a finite morphism. Let x0 ∈ X 0 and let x, s0 , s be its images in X, S 0 , S, respectively. By 2.8.20, Ses¯0 0 is a connected component of S 0 ×S Ses¯ and Ses¯0 0 → Ses¯ is finite. e 0 0 is a connected component of X ex¯ × e Se0 0 . But k(¯ Similarly X s0 ) is a purely x ¯ Ss¯ s¯ ex¯ × e Se0 0 inseparable extension of k(¯ s), so there exists only one point in X

ex¯ . Hence we must have above the closed point of X ex¯0 0 ∼ ex¯ × e Ses¯0 0 . X =X

Ss¯

s¯

Ss¯

For any algebraic geometric point t → Ses¯0 0 , t is also an algebraic geometric point of Ses¯, and we have ex¯ × e t. ex¯0 0 × e0 t ∼ X =X Ss¯0

Ss¯

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If f is locally acyclic relative to K at x, then we have an isomorphism

It follows that

ex¯ × e t, K). Kx¯ ∼ = RΓ(X Ss¯

e 0 0 × e0 t, K 0 ). Kx¯0 0 ∼ = RΓ(X x ¯ S 0 s ¯

0

0

0

So f is locally acyclic relative to K at x . The same argument also shows that if f 0 is locally acyclic relative to K 0 at x0 , then f is locally acyclic relative to K at x. Our assertion follows. Corollary 7.6.4. Let f : X → S be a morphism, A a ring, K ∈ b ob Dtf (X, A), (Sλ , uλµ ) an inverse system of schemes affine and quasi-finite over S, S 0 = limλ Sλ , f 0 : X 0 = X ×S S 0 → S 0 and K 0 = K|X 0 the base ←− changes of f and K, respectively. If f is locally acyclic (resp. strongly locally acyclic) relative to K, then f 0 is locally acyclic (resp. strongly locally acyclic) relative to K 0 . Conversely, if f 0 is locally acyclic (resp. strongly locally acyclic) relative to K 0 and S 0 → S is surjective, then is f locally acyclic (resp. strongly locally acyclic) relative to K. Proof. We prove the statements for local acyclicity. Let x0 ∈ X 0 and let s0 , xλ , sλ , x, s be its images in S 0 , Xλ = X ×S Sλ , Sλ , X, S, respectively. Then we have ∼X ∼ lim X e × Se0 . ex¯ × e Seλ,¯s , X e0 0 = e eλ,¯x ∼ X Ses¯0 0 ∼ Se , X = lim λ λ = x ¯ Ss¯ ←− λ,¯xλ = x¯ Ses¯ s¯0 ←− λ,¯sλ λ

λ

Ses¯0 0

Any algebraic geometric point t of is also an algebraic geometric point of Ses¯. We then use the same argument as in the proof of 7.6.3.

Lemma 7.6.5. Let (A, m) be a normal local integral domain, K its fraction field, and k = A/m. If K is separably closed, then A is strictly local.

Proof. Let K be an algebraic closure of K, let A be the integral closure of A in K, let m be a maximal ideal of A, and let k¯ = A/m. Then m is above the maximal ideal m of A, and hence k¯ is an algebraic extension of k. First let us prove k is separably closed. Let f (t) ∈ k[t] be a monic polynomial and let F (t) ∈ A[t] be a monic polynomial lifting f (t). Then we have F (t) = (t − a1 ) · · · (t − an ) for some a1 , . . . , an ∈ A, and hence f (t) = (t − a ¯1 ) · · · (t − a ¯n ),

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where for each i, a ¯i is the image of ai in k. If a ¯i is a simple root of f (t), then ai is a simple root of F (t). Since K is separably closed, we must have ai ∈ K. Since A is normal, we must have ai ∈ A. So we have a ¯i ∈ k. Thus any simple root of f (t) lies in k, and hence k is separably closed. Next we show that A is henselian. Let F (t) ∈ A[t] be a monic polynomial, f (t) ∈ k[t] the reduction of F (t) mod m, and f (t) = (t − a ¯)g(t) a factorization of f (t) in k[t] such that g(t) is monic and relatively prime to t − a ¯. Again let F (t) = (t − a1 ) · · · (t − an ) be a factorization of F (t) in A[t]. It induces a factorization f (t) = (t − a ¯1 ) · · · (t − a ¯n ) ¯ of f (t) in k[t]. Without loss of generality, assume a ¯ = a ¯1 . Since g(t) is relatively prime to t − a ¯, a ¯1 is a simple root of f (t). As before, this implies that a1 ∈ A. We thus have a factorization F (t) = (t − a1 )G(t) in A[t] lifting the factorization f (t) = (t − a)g(t). By Nakayama’s lemma, the ideal of A[t] generated by t − a1 and G(t) is A[t]. By 2.8.3 (vi), A is henselian. b Lemma 7.6.6. Let f : X → S be a morphism, A a ring, K ∈ ob Dtf (X, A), and γ : t → S an algebraic geometric point. Fix notation by the following Cartesian diagram: γ0

X ×S t → X ft ↓ ↓f γ t → S. (i) If f is locally acyclic relative to K, then we have a canonical isomorphism ∼ =

∗ 0 0∗ K ⊗L A f Rγ∗ A → Rγ∗ γ K.

(ii) If f is strongly locally acyclic relative to K, then for any A-module M , we have a canonical isomorphism ∼ =

∗ 0 0∗ L ∗ K ⊗L A f Rγ∗ M → Rγ∗ (γ K ⊗A ft M ).

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Proof. We prove (ii). Let S 0 be the normalization in t of {γ(t)} with the reduced closed subscheme structure. Fix notation by the following commutative diagram: β0

α0

Xt → X × S S 0 → X ft ↓ f0 ↓ f ↓ β

α

S0

t→

→ S,

where αβ = γ. Note that α : S 0 → S is an integral morphism. Using 5.9.7 and the same argument as the proof of 7.4.7, one can show that K ⊗L f ∗ Rα∗ Rβ∗ M ∼ = Rα0 (α0∗ K ⊗L f 0∗ Rβ∗ M ). A

∗

A

We have ∗ L ∗ ∼ K ⊗L A f Rγ∗ M = K ⊗A f Rα∗ Rβ∗ M 0∗ ∼ = Rα0∗ (α0∗ K ⊗L A f Rβ∗ M ) ∼ Rα0 Rβ 0 (β 0∗ α0∗ K ⊗L f ∗ M ). Rγ 0 (γ 0∗ K ⊗L f ∗ M ) = ∗

A

t

∗

∗

A

t

To prove our assertion, it suffices to show 0∗ 0 0∗ 0∗ L ∗ ∼ α0∗ K ⊗L A f Rβ∗ M = Rβ∗ (β α K ⊗A ft M ).

For any s0 ∈ S 0 , OS 0 ,s0 is a normal local integral domain, and its fraction field is k(t), which is separably closed. By 7.6.5, OS 0 ,s0 is strictly local. So the strict localization Ses¯0 0 of S 0 at s¯0 can be identified with Spec OS 0 ,s0 . It follows that the squares in the following commutative diagram are Cartesian: j t → Se0 0 ← s¯0 s¯

k

β

↓

0

k

t → S ← s¯0 .

So the squares in the following commutative diagram are Cartesian: ˜

˜i j Xt → X ×S Ses¯0 0 ← Xs¯0 k ↓ k β0

i0

Xt → X ×S S 0 ← Xs¯0 .

Let

f˜ : X ×S Ses¯0 0 → Ses¯0 0

be the base change of f . We then have i0∗ (α0∗ K ⊗L f 0∗ Rβ∗ M ) ∼ = ˜i∗ ((K| A

e0 ) X×S S s ¯0

˜∗ ⊗L A f Rj∗ M ),

∗ L ∗ ∼ ˜∗ ˜ ˜∗ i0∗ Rβ∗0 (β 0∗ α0∗ K ⊗L e0 ) ⊗A ft M ). A ft M ) = i Rj∗ (j (K|X×S S s ¯0

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By 7.6.4, f 0 is strongly locally acyclic relative to α0∗ K. So we have L ˜∗ L ∗ ∼ ˜∗ ˜ ˜∗ ˜i∗ ((K| e0 ) ⊗A f Rj∗ M ) = i Rj∗ (j (K|X×S S e0 ) ⊗A ft M ). X×S S s ¯0

s ¯0

It follows that

0∗ L ∗ ∼ 0∗ 0 0∗ 0∗ i0∗ (α0∗ K ⊗L A f Rβ∗ M ) = i Rβ∗ (β α K ⊗A ft M ).

This is true for any s0 ∈ S 0 . So we have 0∗ 0 0∗ 0∗ L ∗ ∼ α0∗ K ⊗L A f Rβ∗ M = Rβ∗ (β α K ⊗A ft M ).

Lemma 7.6.7. Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f g S 0 → S.

b Let A be a noetherian ring and let K ∈ ob Dtf (X, A). Suppose that S is a noetherian scheme, g is an open immersion, and one of the following conditions holds: (a) f is locally acyclic relative to K, and every finitely generated Amodule can be embedded into a free A-module. (b) f is strongly locally acyclic relative to K. Then for any L ∈ ob D+ (S 0 , A), we have a canonical isomorphism ∼ =

∗ 0 0∗ L 0∗ K ⊗L A f Rg∗ L → Rg∗ (g K ⊗A f L).

Remark 7.6.8. Note that if A = R/I for a discrete valuation ring R and a nonzero ideal I of R, then every finitely generated A-module can be embedded into a free A-module. Every finitely generated Z/n-module can also be embedded into a free Z/n-module. Such coefficient rings A are enough for application. Proof.

First we prove that under the condition (b) (resp. (a)), we have K ⊗L f ∗ Rg∗ γ∗ M ∼ = Rg 0 (g 0∗ K ⊗L f 0∗ γ∗ M ) A

∗

A

for any algebraic geometric point γ : t → S 0 and any A-module M (resp. M = A). We have γ∗ M ∼ = Rγ∗ M.

If f is strongly locally acyclic relative to K, then f 0 is strongly locally acyclic relative to g 0∗ K since g is an open immersion. Fix notation by the following commutative diagram: γ0

g0

Xt → X × S S 0 → X ft ↓ f0 ↓ f ↓ γ g t→ S0 → S.

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By 7.6.6, we have isomorphisms ∗ 0 0 0 0 ∗ L ∗ ∼ K ⊗L A f R(gγ)∗ M = R(g γ )∗ ((g γ ) K ⊗A ft M ), g 0∗ K ⊗L f 0∗ Rγ∗ M ∼ = Rγ 0 (γ 0∗ g 0∗ K ⊗L f ∗ M ). A

∗

A

t

It follows that ∗ L ∗ ∼ K ⊗L A f Rg∗ γ∗ M = K ⊗A f Rg∗ Rγ∗ M ∗ ∼ = Rg∗0 Rγ∗0 (γ 0∗ g 0∗ K ⊗L A ft M ) ∼ = Rg 0 (g 0∗ K ⊗L f 0∗ Rγ∗ M ) ∗

A

0∗ ∼ = Rg∗0 (g 0∗ K ⊗L A f γ∗ M ).

This proves our assertion. If f is locally acyclic, the above argument works for M = A. ∗ 0 0∗ L 0∗ ∼ By 6.5.2, to prove K ⊗L A f Rg∗ L = Rg∗ (g K ⊗A f L) for any L ∈ + 0 ob D (S , A), it suffices to treat the case where L = F for a sheaf F of A-modules on S 0 . For convenience, let 0∗ T q (F ) = Rq g∗0 (g 0∗ K ⊗L A f F ),

∗ S q (F ) = H q (K ⊗L A f Rg∗ F ),

and let φqF : S q (F ) → T q (F ) be the homomorphisms induced by the ∗ 0 0∗ L 0∗ canonical morphism K ⊗L A f Rg∗ F → Rg∗ (g K ⊗A f F ). Let us prove q that φF are isomorphisms by induction on q. If K can be represented by a bounded complex of flat sheaves of A-modules such that K i = 0 for any i 6∈ [a, b], then both S q (F ) and T q (F ) vanish for any q < a. So φqF is an isomorphism for any q < a. Suppose we have shown that φqF is an isomorphism for any q < n, and let us prove that φnF is an isomorphism. By 5.8.8 and 5.9.6, it suffices to prove that φnF is an isomorphism for any constructible F . In this case, there exists a decomposition S 0 = ∪λ Uλ of S 0 by finitely many irreducible locally closed subsets such that F |Uλ are locally constant. For each λ, let γλ : tλ → S 0 be an algebraic geometric point lying above the generic point of Uλ , and let Mλ = Ftλ . Note that the canonical morphism F |Uλ → (γλ∗ Mλ )|Uλ is injective. So the canonical morphism Y F → γλ∗ Mλ λ

is injective. Thus we can find a short exact sequence of the form L

0→F →G →H →0

such that G = γ∗ M for some algebraic geometric points γ : t → S 0 and some finitely generated A-modules M , and we can take M = A under the

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condition (a). We have a commutative diagram S n−1 (G ) → S n−1 (H ) → S n (F ) → S n (G ) → S n (H ) ↓ φn φn φn φn−1 ↓ φn−1 F ↓ G ↓ H ↓ G H n−1 n−1 n n n T (G ) → T (H ) → T (F ) → T (G ) → T (H ), where the horizontal lines are exact. By the induction hypothesis, φn−1 G and φn−1 are bijective. By what we have shown at the beginning, φnG is H bijective. These facts imply that φnF is injective for any constructible sheaf F , and hence for any sheaf F . In particular, φnH is injective. These facts imply that φnF is bijective for any constructible sheaf F , and hence for any sheaf F . Theorem 7.6.9. Let A be a noetherian ring, S a noetherian scheme, f : b X → S a morphism, and K ∈ ob Dtf (X, A). The following conditions are equivalent: (i) f is universally strongly locally acyclic relative to K. (ii) For any Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S →S such that g is quasi-compact and quasi-separated, and for any L ∈ ob D+ (S 0 , A), we have a canonical isomorphism ∼ =

∗ 0 0∗ L 0∗ K ⊗L A f Rg∗ L → Rg∗ (g K ⊗A f L).

This remains true if we replace f by fT : X ×S T → T for any base change T → S with T noetherian. Suppose furthermore that every finitely generated A-module can be embedded into a free A-module. Then the above two conditions are equivalent to the following: (iii) f is universally locally acyclic relative to K. Proof. (i)⇒ (ii) The second statement follows from the first one. Let us prove the first statement. The problem is local with respect to S, so we may assume that S is affine. Cover S 0 by finitely many affine open subsets Uα .

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Similar to 5.6.10, we have spectral sequences Y ∗ E1pq = H q K ⊗L A f R(g|Uα0 ×S 0 ···×S 0 Uαp )∗ (L|Uα0 ×S 0 ···×S 0 Uαp ) α0 ,...,αp

∗ ⇒ H p+q (K ⊗L A f Rg∗ L) Y = Rq (g 0 |Uα0 ×S0 ···×S0 Uαp ×S X )∗

E1pq

α0 ,...,αp

0∗ ((g 0∗ K ⊗L A f L)|Uα0 ×S 0 ···×S 0 Uαp ×S X )

0∗ ⇒ Rp+q g∗0 (g 0∗ K ⊗L A f L).

It suffices to prove ∗ K ⊗L A f R(g|Uα0 ×S 0 ···×S 0 Uαp )∗ (L|Uα0 ×S 0 ···×S 0 Uαp )

0∗ ∼ = R(g 0 |Uα0 ×S0 ···×S0 Uαp ×S X )∗ ((g 0∗ K ⊗L A f L)|Uα0 ×S 0 ···×S 0 Uαp ×S X ).

We are thus reduced first to the case where S 0 → S is separated, and then to the case where S = Spec B and S 0 = Spec B 0 are affine. Write B 0 = limλ Bλ , where Bλ goes over the family of subalgebras of B 0 finitely −→ generated over B. Let Sλ = Spec Bλ , and fix notation by the following diagram: π0

By 5.9.6, we have

g0

λ λ X ×S S 0 → X × S Sλ → X f0 ↓ fλ ↓ f ↓ gλ πλ S0 → Sλ → S.

L∼ π ∗ Rπ L. = lim −→ λ λ∗ λ

Hence 0∗ ∼ 0∗ ∗ g 0∗ K ⊗L (g 0∗ K ⊗L A f L = lim A f πλ Rπλ∗ L) −→ λ

0∗ ∗ ∼ (g 0∗ K ⊗L = lim A πλ fλ Rπλ∗ L) −→ λ

∗ ∼ π 0∗ (g 0∗ K ⊗L = lim A fλ Rπλ∗ L). −→ λ λ λ

By 5.9.6 again, we have 0∗ ∗ ∼ Rg∗0 (g 0∗ K ⊗L Rg 0 (g 0∗ K ⊗L A f L) = lim A fλ Rπλ∗ L). −→ λ∗ λ λ

If (ii) holds in the case where g is affine of finite type, then we have ∗ L ∗ 0 0∗ L ∗ ∼ ∼ K ⊗L A f Rg∗ L = K ⊗A f Rgλ∗ Rπλ∗ L = Rgλ∗ (gλ K ⊗A fλ Rπλ∗ L)

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for all λ, and hence ∗ 0 0∗ L 0∗ ∼ K ⊗L A f Rg∗ L = Rg∗ (g K ⊗A f L).

We are thus reduced to the case where B 0 is a finitely generated B-algebra. We then have an epimorphism B[T1 , . . . , Tn ] → B 0 . It defines a closed immersion i : S 0 → AnS = Spec B[T1 , . . . , Tn ]. Let j : AnS ,→ PnS = Proj B[T0 , . . . , Tn ] be the canonical open immersion and let π : PnS → S be the projection. We have g = πji. Fix notation by the following commutative diagram of Cartesian squares: i0

j0

π0

i

j

π

X ×S S 0 → AnX → PnX → X f0 ↓ f1 ↓ f2 ↓ f ↓ S0

→ AnS → PnS → S.

Since π and i are proper, by 7.4.8, we have ∗ 0 0∗ L ∗ ∼ K ⊗L A f Rπ∗ Rj∗ Ri∗ L = Rπ∗ (π K ⊗A f2 Rj∗ Ri∗ L), ∗ L 0∗ ∼ 0 0∗ 0∗ 0∗ j 0∗ π 0∗ K ⊗L A f1 Ri∗ L = Ri∗ (i j π K ⊗A f L).

Since f is universally strongly locally acyclic relative to K, f2 is strongly locally acyclic relative to π 0∗ K. Applying 7.6.7 to the middle Cartesian square in the above diagram, we get ∗ 0 0∗ 0∗ L ∗ ∼ π 0∗ K ⊗L A f2 Rj∗ Ri∗ L = Rj∗ (j π K ⊗A f1 Ri∗ L).

So we have ∗ L ∗ ∼ K ⊗L A f Rg∗ L = K ⊗A f Rπ∗ Rj∗ Ri∗ L ∼ = Rπ 0 (π 0∗ K ⊗L f ∗ Rj∗ Ri∗ L) ∗

A

2

∗ ∼ = Rπ∗0 Rj∗0 (j 0∗ π 0∗ K ⊗L A f1 Ri∗ L) ∼ = Rπ 0 Rj 0 Ri0 (i0∗ j 0∗ π 0∗ K ⊗L f 0∗ L) ∗

∗

∗

A

0∗ ∼ = Rg∗0 (g 0∗ K ⊗L A f L).

This proves (ii). (ii)⇒(i) Let S 0 → S be a base change with S 0 being noetherian and let s0 ∈ S 0 . Then the strict localization Ses¯0 0 of S 0 at s¯0 is noetherian. For

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any algebraic geometric point γ : t → Ses¯0 0 , applying (ii) to the Cartesian diagram γ0

X ×S t → X ×S Ses¯0 0 ft ↓ ↓ f˜ γ t → Ses¯0 0 ,

we get a canonical isomorphism

0 0∗ L ∗ ˜∗ ∼ K|X×S Se0 ⊗L e0 0 ) ⊗A ft M ) A f Rγ∗ M = Rγ∗ (γ (K|X×S S 0 s ¯

s ¯

for any A-module M . Taking the restriction to the closed fiber Xs¯0 of X ×S Ses¯0 0 , we get 0 0∗ L ∼ K|Xs¯0 ⊗L e0 ) ⊗A M ))|Xs¯0 . A M = (Rγ∗ (γ (K|X×S S 0 s ¯

0

0

Hence X ×S S → S is strongly locally acyclic relative to K|X×S S 0 . Now suppose that every finitely generated A-module can be embedded into a free A-module. (i)⇒(iii) is trivial. (iii)⇒(ii) can be proved by the same argument as (i)⇒(ii). Suppose that f : X → S is locally acyclic relative to K ∈ ob D+ (X). For any q, any s ∈ S and any algebraic geometric point j : t → Ses¯, we define the cospecialization homomorphism to be the composite e H q (Xt , K|Xt ) ∼ = H q (X ×S Ses¯, R˜j∗˜j ∗ K) e → H q (Xs¯, ˜i∗ R˜j∗ ˜j ∗ K) ∼ = H q (Xs¯, K|Xs¯ ).

Proposition 7.6.10. Let A be a noetherian ring, S a noetherian scheme, and f : X → S a morphism locally acyclic relative to A. Suppose that every finitely generated A-module can be embedded into a free A-module, and for any s ∈ S and any geometric point j : t → Ses¯, cospecialization homomorphisms H q (Xt , A) → H q (Xs¯, A)

are bijective for all q. Then for any sheaf F of A-modules on S, the canonical morphism (Rf∗ f ∗ F )s¯ → RΓ(Xs¯, (f ∗ F )|Xs¯ ) is an isomorphism.

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Proof.

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It suffices to prove that the canonical morphism f)s¯ → RΓ(Xs¯, (f˜∗ F f)|X ) (Rf˜∗ f˜∗ F s ¯

f on Ses¯. We have is an isomorphism for any sheaf of A-modules F f˜∗ j∗ A ∼ = f˜∗ Rj∗ A ∼ = R˜j∗ A

by 7.6.6. Since cospecialization homomorphisms are isomorphisms, the canonical morphism RΓ(X ×S Ses¯, R˜j∗ A) → RΓ(Xs¯, ˜i∗ R˜j∗ A)

is an isomorphism. So the canonical morphism

f)s¯ → RΓ(Xs¯, (f˜∗ F f)|X ) (Rf˜∗ f˜∗ F s ¯

f = j∗ A. Since any sheaf of A-modules on Ses¯ is is an isomorphism for F a direct limit of constructible sheaves of A-modules, and any constructible L sheaf of A-modules can be embedded into a sheaf of the form j j∗ A, we have f)|X ) f)s¯ ∼ (Rf˜∗ f˜∗ F = RΓ(Xs¯, (f˜∗ F s ¯

f on Ses¯. (Confer the proof of 7.6.7.) for any sheaf of A-modules F

7.7

Smooth Base Change Theorem

([SGA 4] XVI, [SGA 4 21 ] Arcata V 2, 3) The main result of this section is the following. Theorem 7.7.1. Let f : X → S be a smooth morphism, and let n be an integer invertible on S. Then f is universally locally acyclic relative to Z/n. Before proving this theorem, we give several applications. Theorem 7.7.2 (Smooth base change theorem). Consider a Cartesian diagram g0

X ×S S 0 → X f0 ↓ ↓f 0 g S → S.

If f is smooth and g is quasi-compact and quasi-separated, then for any integer n invertible on S and any L ∈ D+ (S 0 , Z/n), we have f ∗ Rg∗ L ∼ = Rg∗0 f 0∗ L.

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Proof. The problem is local with respect to S and X. We may assume that X and S are affine. As in the beginning of the proof of 7.6.9, we may assume that S 0 is also affine. We can find an inverse system of noetherian affine schemes {Sλ } such that S = limλ Sλ and f : X → S can be descended ←− down to an inverse system of smooth morphisms fλ : Xλ → Sλ . Fix notation by the following commutative diagram of Cartesian squares: u0

g0

λ X ×S S 0 → X → Xλ f0 ↓ ↓f ↓ fλ g uλ S0 → S → Sλ . By 7.7.1, each fλ is universally locally acyclic relative to Z/n. Note that every finitely generated Z/n-module can be embedded into a free Z/nmodule. By 7.6.9, we have fλ∗ R(uλ g)∗ L ∼ = R(u0λ g 0 )∗ f 0∗ L. By 5.9.6, we have ∼ f ∗ (lim u∗ R(uλ g)∗ L) f ∗ Rg∗ L = −→ λ

λ

∼ f ∗ u∗λ R(uλ g)∗ L = lim −→ λ

∼ u0∗ f ∗ R(uλ g)∗ L, = lim −→ λ λ λ

Rg∗0 f 0∗ L ∼ u0∗ R(u0λ g 0 )∗ f 0∗ L. = lim −→ λ λ It follows that f ∗ Rg∗ L ∼ = Rg∗0 f 0∗ L.

Corollary 7.7.3. Let K/k be an extension of a separably closed field, X a quasi-compact quasi-separated k-scheme, n an integer relatively prime to the characteristic of k, and F a sheaf of Z/n-modules on X. Then the canonical homomorphisms H q (X, F ) → H q (X ⊗k K, F |X⊗k K ) are isomorphisms. Proof. By 5.7.2 (ii) and 5.9.2, we may assume that k and K are algebraically closed. We have K = limλ Kλ , where Kλ are subfields of K −→ finitely generated over k. For each λ, choose a separating transcendental basis {x1 , . . . , xnλ } for Kλ over k. Then Kλ is finite and separable over k(x1 , . . . , xnλ ). Fix notation by the following commutative diagram e0

g0

h0

g0

λ λ U U X ⊗k K → X ⊗k Kλ → X ⊗k k(x1 , . . . , xnλ ) → X ×k U → 0 0 f ↓ fλ ↓ fλ ↓ fU ↓

e

gλ

h

λ U Spec K → Spec Kλ → Spec k(x1 , . . . , xnλ ) →

U

gU

X ↓

f

→ Spec k,

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where U goes over the set of nonempty affine open subsets of Spec k[x1 , . . . , xnλ ]. By the smooth base change theorem 7.7.2, we have ∼ Rq fU∗ g 0∗ F . g ∗ R q f∗ F = U

U

By 5.9.6, we have 0∗ 0∗ ∼ Rq fλ∗ h0∗ h∗ Rq fU∗ gU F. U gU F = lim −→ U U

Since gλ is etale, we have 0∗ ∼ q 0 0∗ 0∗ 0∗ gλ∗ Rq fλ∗ h0∗ U gU F = R fλ∗ gλ hU gU F .

Denote by g the morphism Spec K → Spec k and by g 0 : X ⊗k K → X its base change. By 5.9.6, we have ∼ lim e∗ Rq f 0 g 0∗ h0∗ g 0∗ F Rq f∗0 g 0∗ F = λ∗ λ U U −→ λ λ

0∗ ∼ e∗ g ∗ Rq fλ∗ h0∗ = lim U gU F −→ λ λ λ

0∗ ∼ lim e∗ g ∗ h∗ Rq fU∗ gU F = lim −→ −→ λ λ U λ

U

∼ lim e∗ g ∗ h∗ g ∗ Rq f∗ F = lim −→ −→ λ λ U U λ

U

∼ = g ∗ R q f∗ F .

Since k and K are algebraically closed, this is equivalent to saying that ∼ H q (X, F ). H q (X ⊗k K, F |X⊗ K ) = k

Corollary 7.7.4. Let S be a scheme, π : AkS = Spec OS [T1 , . . . , Tk ] → S the canonical morphism, n an integer invertible on S, and K ∈ ob D+ (S, Z/n). Then the canonical morphism K → Rπ∗ π ∗ K is an isomorphism. Proof. It suffices to treat the case where k = 1 by applying this special case to each projection in the sequence AkS → Ak−1 → · · · → A1S → S. S For any s ∈ S, let k(s) be a separable closure of the residue field k(s). We have Pic(A1k(s) ) = 0 since k(s)[T ] is a unique factorization domain. By 7.2.9 (i), we have 1 , Z/n) ∼ H q (Ak(s) = H q (A1k(s) , µn ) ∼ =

Z/n 0

if q = 0, if q ≥ 1.

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By 7.7.1, π is universally locally acyclic relative to Z/n. The cospecialization homomorphisms for Z/n are isomorphisms. By 7.6.10, for any sheaf of Z/n-modules F on S, we have an isomorphism (Rπ∗ π ∗ F )s¯ → RΓ(A1k(s) , π ∗ F |A1

k(s)

But 1 , π ∗ F |A 1 H q (Ak(s)

k(s)

)∼ =

Fs¯ 0

).

if q = 0, if q ≥ 1.

It follows that F ∼ = Rπ∗ π ∗ F . Hence K∼ = Rπ∗ π ∗ K for any K ∈ ob D+ (S, Z/n).

Corollary 7.7.5. Let f : X → S be a smooth morphism and let n be an integer invertible on S. Then for any locally constant sheaf of Z/n-modules F on X, f is universally strongly locally acyclic relative to F . Proof. By 7.5.7 and 7.6.2, it suffices to show that f is universally locally acyclic relative to F . Since base changes of smooth morphisms are smooth, it suffices to prove that f is locally acyclic relative to F . For any s ∈ S, and any algebraic geometric point j : t → Ses¯, fix notation by the following diagram: j0

Vt = V ×Ses¯ t → pt ↓ ˜

V p↓

i0

← Vs¯ = V ×Ses¯ s¯ ↓ ps¯

˜i j Xt = X ×S t → X ×S Ses¯ ← Xs¯ = X ×S s¯ ft ↓ f˜ ↓ ↓ fs¯ j i e t→ Ss¯ ← s¯.

f= where p : V → X ×S Ses¯ is a morphism that will be used later. Set F F |X×S Ses¯ . We need to show that the canonical morphism f f → ˜i∗ R˜j∗ ˜j ∗ F ˜i∗ F

is an isomorphism. For any x ∈ Xs¯, we need to check f)x¯ f)x¯ → (˜i∗ R˜j∗ ˜j ∗ F (˜i∗ F

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is an isomorphism. Since F is locally constant, we can find an etale morf|V is a phism p : V → X ×S Ses¯ whose image contains the point x so that F constant sheaf, say associated to a Z/n-module M . It suffices to show that f → p∗˜i∗ R˜j∗˜j ∗ F f p∗s¯˜i∗ F s¯

is an isomorphism. We have

f∼ p∗s¯˜i∗ F = i0∗ M, f∼ f p∗s¯˜i∗ R˜j∗ ˜j ∗ F = i0∗ p∗ R˜j∗ ˜j ∗ F f ∼ = i0∗ Rj 0 p∗˜j ∗ F ∗ t

∼ = i0∗ Rj∗0 M.

So it suffices to show that the canonical morphism i0∗ M → i0∗ Rj∗0 M is an isomorphism. Applying the smooth base change theorem 7.7.2 to the smooth morphism f˜p, we get isomorphisms (f˜p)∗ Rj∗ M ∼ = Rj∗0 (ft pt )∗ M ∼ = Rj∗0 M. So we have (f˜s¯ps¯)∗ i∗ Rj∗ M ∼ = i0∗ (f˜p)∗ Rj∗ M ∼ = i0∗ Rj∗0 M. Our assertion then follows from the fact that M∼ = i∗ Rj∗ M.

We now prepare to prove 7.7.1. f1

f2

Lemma 7.7.6. Let X → Y → S be morphisms of noetherian schemes, b let A be a noetherian ring, and let K ∈ ob Dtf (X, A). Suppose that f1 is universally strongly locally acyclic relative to K, and f2 is universally strongly locally acyclic relative to A. Then f2 f1 is universally strongly locally acyclic relative to K. If either both f1 and f2 are of finite type, or any finitely generated A-module can be embedded into a free A-module, then we have the same conclusion under the assumption that f1 is universally locally acyclic relative to K, and f2 is universally locally acyclic relative to A.

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Proof. The second part follows from the first part by 7.6.2 and 7.6.9. Consider a commutative diagram of Cartesian squares g0

X ×S S 0 → X f10 ↓ ↓ f1 g00

Y ×S S 0 → Y f20 ↓ ↓ f2 g S0 → S

such that g is quasi-compact and quasi-separated. Since f1 is universally strongly locally acyclic relative to K and f2 is universally strongly locally acyclic relative to A, we have 0∗ L 0∗ 0∗ ∗ 00 0∗ ∼ 0 K ⊗L A f1 Rg∗ f2 L = Rg∗ (g K ⊗A f1 f2 L), f2∗ Rg∗ L ∼ = Rg∗00 f20∗ L

for any L ∈ ob D+ (S 0 , A) by 7.6.9. So we have 00 0∗ ∗ ∗ L ∗ ∼ K ⊗L A f1 f2 Rg∗ L = K ⊗A f1 Rg∗ f2 L ∼ = Rg 0 (g 0∗ K ⊗L f 0∗ f 0∗ L). ∗

A

1

2

This remains true if we replace f1 and f2 by their base changes with respect to any T → S with T noetherian. So f2 f1 is universally strongly locally acyclic relative to K by 7.6.9. Lemma 7.7.7. Let (A, m) be a local ring, k = A/m, and K a finite extension of k. Then there exists a local ring (B, n) and a local homomorphism A → B such that B is finite and faithfully flat over A, and B/n is kisomorphic to K. Proof. We may reduce to the case where K is a simple extension of k. Then K is k-isomorphic to k[t]/(f (t)) for some monic irreducible polynomial f (t). Let F (t) ∈ A[t] be a monic polynomial whose reduction mod m is f (t) and let B = A[t]/(F (t)). Then B is finite and faithfully flat over A. Moreover, we have B/mB ∼ = k[t]/(f (t)) ∼ = K. So mB is a maximal ideal of B. Since B is finite over A, any maximal ideal of B is above the unique maximal ideal m of A. So any maximal ideal of B contains mB. It follows that mB is the only maximal ideal of B. So B is local and its residue field is k-isomorphic to K.

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Let f : X → S be a smooth morphism. Let us prove that f is universally locally acyclic relative to Z/n, where n is an integer invertible on S. The problem is local on S and on X. We may assume that S = Spec A is affine. We have A = limλ Aλ , where Aλ are subrings of A finitely generated over −→ Z. We may assume that f can be descended down to a smooth morphism fλ : Xλ → Sλ for some λ. It suffices to prove that fλ is universally locally acyclic relative to Z/n. We are thus reduced to the case where S = Spec A for some finitely generated Z-algebra A. We may assume that f can be factorized as j

X → AkS → S,

where j is an etale morphism. It suffices to prove that AkS → S is universally locally acyclic relative to Z/n. By induction on k and 7.7.6, we are reduced to the case where k = 1. Let s ∈ S and let j : t → Ses¯ be an algebraic geometric point. Fix notation by the following diagram: ˜ j

˜i

A1k(t) → A1Se ← A1k(¯s) s ¯ ↓ ↓ ↓ j i t → Ses¯ ← s¯.

Let us prove that the canonical morphism Z/n → ˜i∗ R˜j∗ Z/n

is an isomorphism, and note that this remains to be true if S is replaced by a base S 0 of finite type over S. It suffices to prove that for any Zariski closed point x in A1k(¯s) , we have Z/n ∼ = (˜i∗ R˜j∗ Z/n)x¯ , that is, e1 H q (A e S

s ¯

e1 where A e S

x s ¯,¯

A1k(¯s)

×Ses¯ t, Z/n) ∼ = ,¯ x

Z/n Z/n

if q = 0, if q ≥ 1,

is the strict localization of A1Se at x¯. The residue field k(x) of s ¯

x in is finite over k(¯ s). By 7.7.7, there exists a local ring (B, n) finite and faithfully flat over OSes¯,¯s such that B/n is k(¯ s)-isomorphic to k(x). B is strictly henselian and isomorphic to the strict henselization at a point of a scheme of finite type over Z. Replacing k(t) by a purely inseparable algebraic extension, we may assume that the algebraic geometric point j : t → Ses¯ can be lifted to an algebraic geometric point t → Spec B. The k(¯ s)-isomorphism k(x) ∼ = B/n defines an s¯-morphism Spec B/n → A1k(¯s)

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with image x. The composite Spec B/n → A1k(¯s) → A1Se

s ¯

and the morphism Spec B/n → Spec B induce a morphism Spec B/n → A1Se ×Ses¯ Spec B = A1B . s ¯

A1B

Let y ∈ be the image of this morphism. Then y lies above x, and k(y) ∼ = B/n. Note that B/n is a separably closed field and finite purely inseparable over the separable closed field k(¯ s). By 2.8.20, we have e1 A e S

x s ¯,¯

Hence

e1 A e S

x s ¯,¯

e1 . ×Ses¯ Spec B ∼ =A B,¯ y

e 1 ×Spec B t. ×Ses¯ t ∼ =A B,¯ y

We are thus reduced to proving

e 1 ×Spec B t, Z/n) ∼ H q (A = B,¯ y

Z/n 0

if q = 0, if q ≥ 1.

Since y ∈ A1B lies above the closed point of Spec B, it lies in the closed subscheme Spec (B/n)[T ] of A1B = Spec B[T ]. Since k(y) ∼ = B/n, y corresponds to a prime ideal of (B/n)[T ] of the form (T − ¯b) for some ¯b ∈ B/n. Let b ∈ B be a lifting of ¯b. Then y corresponds to the prime ideal of B[T ] generated by n and T − b. Making the change of variable T 7→ T − b, we may assume that y corresponds to the prime ideal of B[T ] generated by n and T . We are thus reduced to proving the following lemma. Lemma 7.7.8. Let (B, n) be the strict henselization at a point of a scheme of finite type over Z, n an integer invertible in B, S = Spec B, B{T } the strict henselization of B[T ] at the prime ideal generated by n and T , X = Spec B{T }, and j : t → S an algebraic geometric point. Then Z/n if q = 0, H q (Xt , Z/n) ∼ = 0 if q ≥ 1. Proof. Let S 0 be the normalization of {j(t)} with the reduced closed subscheme structure. By [Matsumura (1970)] (31.H) Theorem 72, S 0 is finite over S, and hence is strictly local. It is also the strict localization at a point of a scheme of finite type over Z. Replacing S by S 0 , we are reduced

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to the case where S is normal and t lies above the generic point of S. Then X is also normal. Xt is the inverse limit of an inverse system of smooth affine curves over k(t). By 5.9.3 and 7.2.10, we have H q (Xt , Z/n) = 0 for any q ≥ 2. It remains to prove H 0 (Xt , Z/n) ∼ = Z/n,

H 1 (Xt , Z/n) = 0.

As Z/n ∼ = µn,Xt on Xt , it suffices to prove ∼ Z/n, H 1 (Xt , µn,X ) = 0. H 0 (Xt , Z/n) = t We have k(t) = limL L, where L goes over the family of subfields of k(t) −→ finite over the fraction field of B. By 5.9.3, it suffices to prove lim H 0 (XL , Z/n) ∼ H 1 (XL , µn,XL ) = 0, = Z/n, lim −→ −→ L

L

where XL = X ⊗B L. By Kummer’s theory 7.2.1, we have a short exact sequence ∗ ∗ )n → H 1 (XL , µn,XL ) )/Γ(XL , OX 0 → Γ(XL , OX L L

→ Ker (n : Pic(XL ) → Pic(XL )) → 0.

For each L, let BL be the integral closure of B in L. It is also the strict henselization at a point of a scheme of finite type over Z. We have XL ∼ = Spec (BL {T } ⊗B L). L

It suffices to prove the following: (i) XL is connected and nonempty for each L. (ii) limL (BL {T } ⊗BL L)∗ /((BL {T } ⊗BL L)∗ )n = 0, where (BL {T } ⊗BL −→ L)∗ is the group of units of BL {T } ⊗BL L. (iii) Ker (n : Pic(XL ) → Pic(XL )) = 0 for each L. Note that if A is a strictly henselian local ring, then A[T ]/T mA[T ] and A{T }/T mA{T } are strictly henselian. We have A{T }/T mA{T } ∼ = (A[T ]/T mA[T ]) ⊗A[T ] A{T }.

By 2.8.20, A{T }/T mA{T } is isomorphic to the strict henselization of A[T ]/T mA[T ]. It follows that A[T ]/T mA[T ] ∼ = A{T }/T mA{T },

and hence lim A{T }/T mA{T } ∼ = A[[T ]]. ←− m

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Since A{T } is noetherian and T lies in the maximal ideal of A{T }, the canonical homomorphism A{T } → lim A{T }/T mA{T } ← − m

is injective. If A is an integral domain, then so are A[[T ]] and A{T }. Since BL is an integral domain, BL {T } and hence BL {T } ⊗B L are integral domains. In particular, XL = Spec (BL {T } ⊗B L) is connected. It is clear that BL {T } ⊗B L is a nontrivial ring. So XL is nonempty. This proves (i). To prove (ii), it suffices to show every unit of BL {T } ⊗BL L is a product of a unit in BL {T } and a nonzero element in L. This is because every unit of the strictly local ring BL {T } has an n-th root by 2.8.3 (v), and every √ nonzero element α in L has an n-th root in L[ n α] ⊂ k(t). Let f be a unit of BL {T } ⊗BL L. Then f is a product of an element in BL {T } and a nonzero element in L. To prove that f is a product of a unit in BL {T } and a nonzero element in L, it suffices to consider the case where f ∈ BL {T } ∩ (BL {T } ⊗BL L)∗ . Let V be the subset of Spec BL of regular points. Since BL is the strict henselization at a point of a scheme of finite type over Z, V is an open subset of Spec BL ([Matsumura (1970)] Chapter 13). Since BL is normal, V contains all the prime ideals of BL of height 1. Let U be the inverse image of V under the canonical morphism πL : Spec BL {T } → Spec BL .

Then U is regular. Let q be a prime ideal of BL {T } and let p = q ∩ BL . By [Matsumura (1970)] (13.B) Theorem 19 (2), we have ht q = ht p + ht(q/pBL {T }).

In particular, we have ht q ≥ ht p. If ht q = 1, then ht p ≤ 1, and hence p lies in V . So U contains all prime ideals of BL {T } of height 1. Let (f ) be the principle Weil divisor on U defined by f ∈ BL {T }, and let q be a prime ideal of BL {T } of height 1 such that vq (f ) 6= 0, where vq is the valuation defined by q. As f is a unit in BL {T } ⊗BL L, q does not lie in the generic fiber of πL : Spec BL {T } → Spec BL . So p 6= 0, and hence ht p = 1. Note that pBL {T } is a prime ideal of BL {T } since ∼ (BL /p){T } BL {T }/pBL{T } = and (BL /p){T } is an integral domain. So we must have q = pBL {T }. So the divisor (f ) is of the form X (f ) = np{pBL {T }}. ht p=1

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P Let D be the Weil divisor ht p=1 np {p} on V , and let L (D) be the in∗ vertible OV -module corresponding to D. Then πL L (D) is isomorphic to the invertible OU -module corresponding to the divisor (f ). So we have ∗ πL L (D) ∼ = OU . Let iL : Spec BL → Spec BL {T } be the morphism corresponding to the BL -algebra homomorphism BL {T } → BL ,

T 7→ 0.

We have πL ◦ iL = id. It follows that ∗ L (D) ∼ L (D) ∼ = i∗L πL = i∗L OU ∼ = OV .

So D is a principle divisor. Let f0 ∈ L∗ such that D = (f0 ). For any prime ideal q of BL {T } of height 1, if p = q ∩ BL 6= 0, then we have vq (f ) = vq (f0 ) = np . If q ∩ BL = 0, we have vq (f ) = vq (f0 ) = 0. So vq (f ) = vq (f0 ) for all prime ideals q of BL {T } of height 1. It follows that \ f f0 , ∈ (BL {T })q. f0 f ht q=1

Since BL {T } is normal, we have BL {T } =

\

(BL {T })q

ht q=1

by [Matsumura (1970)] (17.H) Theorem 38. So

f f0 f0 , f

∈ BL {T }, and hence

is a unit in BL {T }. We have f = ff0 · f0 and f0 ∈ L∗ . This proves (ii). To prove (iii), let L be an invertible OXL -module satisfying L n ∼ = OXL . We need to show L ∼ = OXL . We have f f0

XL ∼ = U ×V Spec L. Since every Weil divisor on XL can be extended to a Weil divisor on U , the invertible OXL -module L can be extended to an invertible OU -module ∗ ∗ LU . Replacing LU by LU ⊗ πL iL (LU−1 ), we may assume i∗L LU ∼ = OV .

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Let DU be a divisor on U defining LUn . Since L n ∼ = OXL , the restriction DU |XL of DU to XL is a principle divisor. So DU |XL = (f )|XL for some f in the fraction field of BL {T }. Replacing DU by DU − (f ), we may assume that LUn is defined by a divisor DU satisfying DU |XL = 0. Using the argument in the proof of (ii), we see that DU is of the form X DU = np {pBL {T }}. ht p=1

Let MV be the invertible OV -module defined by the divisor Then

P

ht p=1

np{p}.

∗ πL MV ∼ = LUn .

Since i∗L LU ∼ = OV , we have ∗ MV ∼ MV ∼ = i∗L πL = i∗L LUn ∼ = OV .

So LUn ∼ = OU . Let us prove that the conditions i∗L LU ∼ = OV and LUn ∼ = OU imply that ∼ LU = OU . For each nonnegative integer m, let im : Spec(BL {T }/T m+1BL {T }) → Spec BL {T } be the canonical morphism and let Um = i−1 m (U ). We have seen that BL {T }/T m+1BL {T } ∼ = BL [T ]/T m+1BL [T ]. In particular, BL {T }/T m+1BL {T } is a free BL -module of finite rank. Let πm : Spec(BL {T }/T m+1BL {T }) → Spec BL be the canonical morphism. We have Γ(Um , OUm ) ∼ = Γ(V, πm∗ OSpec(BL {T }/T m+1 BL {T }) ) ∼ = Γ(V, (BL {T }/T m+1BL {T })∼). Since BL is normal, we have BL =

\

(BL )p

ht p=1

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by [Matsumura (1970)] (17.H) Theorem 38. Since V contains all prime ideals of BL of height 1, we have Γ(V, OSpec B ) ∼ = Γ(Spec BL , OSpec B ). L

Since (BL {T }/T also have

m+1

∼

BL {T })

L

is a free OSpec BL -module of finite rank, we

Γ(V, (BL {T }/T m+1BL {T })∼ ) ∼ = Γ(Spec BL , (BL {T }/T m+1BL {T })∼) ∼ BL {T }/T m+1BL {T }. = We thus have Γ(Um , OUm ) ∼ = BL {T }/T m+1BL {T }. Hence H 0 (Um , OU∗ m ) ∼ = (BL {T }/T m+1BL {T })∗.

As BL {T }/T m+1BL {T } is strictly henselian, units in BL {T }/T m+1BL {T } have n-th roots. Using the long exact sequence of cohomology groups associated to the Kummer short exact sequence in 7.2.1, we see that H 1 (Um , µn,U ) ∼ = ker(n : Pic(Um ) → Pic(Um )). m

On the other hand, U contains all prime ideals of BL {T } of height 1, and BL {T } is normal. So we have Γ(U, OU ) = BL {T },

H 0 (U, OU∗ ) = BL {T }∗.

As BL {T } is strictly henselian, all units in BL {T } have n-th roots. It follows that H 1 (U, µn,U ) ∼ = ker(n : Pic(U ) → Pic(U )). Note that i0 : Spec(BL {T }/T BL{T }) → Spec BL {T } can be identified with iL : Spec BL → Spec BL {T }. As LU is an element in ker(n : Pic(U ) → Pic(U )) satisfying i∗0 LU ∼ = OU0 , it corresponds to an element c ∈ H 1 (U, µn,U ) whose image under the canonical homomorphism H 1 (U, µn,U ) → H 1 (U0 , µn,U0 )

is 0. Since H 1 (Um , µn,Um ) ∼ = H 1 (U0 , µn,U0 ) by 5.7.2 (i), the image of c under the canonical homomorphism H 1 (U, µn,U ) → H 1 (Um , µn,Um ) is also 0. To prove LU ∼ = OU , it suffices to prove c = 0. Taking into account of 5.7.19, this follows from 7.7.9 below.

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Lemma 7.7.9. Let B be a normal strictly henselian noetherian local ring, V an open subset of Spec B containing all prime ideals of height 1, U the inverse image of V under the canonical morphism π : Spec B{T } → Spec B, Um the inverse images of U under the canonical closed immersions Spec B[T ]/T m+1B[T ] → Spec B{T }, and f : U 0 → U an etale covering 0 space. If the morphisms fm : Um → Um induced by f by base change are trivial etale covering spaces for all m, then f : U 0 → U is a trivial etale covering space. Proof. Let j : U ,→ Spec B{T } be the open immersion. By [Fu (2006)] 1.4.9 (iii), or [Hartshorne (1977)] II 5.8, or [EGA] I 9.2.1, j∗ f∗ OU 0 is a quasi-coherent OSpecB{T } -module. Let C = Γ(Spec B{T }, j∗ f∗ OU 0 ) = Γ(U 0 , OU 0 ). We then have j∗ f∗ OU 0 ∼ = C∼. Since f is affine, we have U0 ∼ = Spec j ∗ j∗ f∗ OU 0 . = Spec f∗ OU 0 ∼ It follows that U0 ∼ = U ×Spec B{T } Spec C. To prove that f : U 0 → U is a trivial etale covering space, it suffices to prove that Spec C → Spec B{T } is a trivial etale covering space. Since B{T } is strictly henselian, it suffices to prove that C is finite and etale over B{T }. As B[[T ]] is faithfully flat over B{T }, it suffices to prove that C ⊗B{T } B[[T ]] is finite and etale over B[[T ]]. Fix notation by the following commutative diagram, where all squares are Cartesian: fm

0 Um → Um ,→ Spec B[T ]/T m+1B[T ] ↓ ↓ ↓ ˆ

ˆ

f

j

j f b0 → b ,→ U U g0 ↓ ↓

U 0 → U ,→ ↓ V ,→

Spec B[[T ]] ↓g Spec B{T } ↓ Spec B.

Since g : Spec B[[T ]] → Spec B{T } is flat, we have

g ∗ j∗ f∗ OU 0 ∼ = ˆj∗ fˆ∗ g 0∗ OU 0 ∼ = ˆj∗ fˆ∗ OUb 0

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by [Fu (2006)] 2.4.10, or [Hartshorne (1977)] III 9.3, or [EGA] III 1.4.15. Hence g∗C ∼ ∼ = ˆj∗ fˆ∗ OUb 0 . We thus have Γ(Spec B[[T ]], g ∗ C ∼ ) ∼ = Γ(Spec B[[T ]], ˆj∗ fˆ∗ OUb 0 ), that is,

Similarly, we have

b 0 , O b 0 ). C ⊗B{T } B[[T ]] ∼ = Γ(U U b, Ob) ∼ Γ(U U = B[[T ]].

Since fm are trivial etale covering spaces, we have canonical isomorphisms 0 0 ) ∼ Γ(Um , OUm = Γ(Um , OUm )k ,

where k is the degree of the finite etale morphism f . The canonical ho0 b 0 , O b 0 ) → Γ(Um 0 ) thus induce a homomorphism of momorphisms Γ(U , OUm U B[[T ]]-algebras b 0 , O b 0 ) → lim Γ(Um , OUm )k . Γ(U U ←− k

Let h : Spec B[T ]/T that

m+1

B[T ] → Spec B be the canonical morphism. Note

h∗ OSpec B[T ]/T m+1 B[T ] ∼ = (B[T ]/T m+1B[T ])∼ is a free OSpec B -module. Since B is normal and V contains all prime ideals of height 1, we have Γ(V, h∗ OSpec B[T ]/T m+1 B[T ] ) ∼ = Γ(Spec B, h∗ OSpec B[T ]/T m+1 B[T ] ) by [Matsumura (1970)] (17.H) Theorem 38. So we have Γ(Um , OUm ) ∼ = B[T ]/T m+1B[T ]. We thus have a homomorphism of B[[T ]]-algebras b 0 , O b 0 ) → lim(B[T ]/T m+1B[T ])k = B[[T ]]k . Γ(U U ←− k

The sheaves ˆj∗ fˆ∗ OUb 0 and ˆj∗ OUb are quasi-coherent OSpec B[[T ]] -modules. We have b 0 , O b 0 )∼ , ˆj∗ fˆ∗ O b 0 ∼ ˆ ˆ b 0 )∼ ∼ = Γ(U U = Γ(Spec B[[T ]], j∗ f∗ OU U b , O b )∼ . ˆj∗ O b ∼ = Γ(U = Γ(Spec B[[T ]], ˆj∗ O b )∼ ∼ U

U

U

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b 0 , O b 0 ) → B[[T ]]k constructed The homomorphism of B[[T ]]-algebras Γ(U U above thus defines a morphism of OSpec B[[T ]] -algebras ˆj∗ fˆ∗ O b 0 → (ˆj∗ O b )k . U

U

b of Spec B[[T ]], we get a Taking its restriction to the open subscheme U morphism of OUb -algebras fˆ∗ O b 0 → O k . b U

U

Since fˆ is affine, we have

b0 ∼ U = Spec fˆ∗ OUb 0 .

b0 → U b, The above morphism thus defines k sections of the morphism fˆ : U and these sections are different since they induce different sections for fm : 0 Um → Um . It follows that fˆ is a trivial etale covering space and b 0, O b0 ) ∼ b , O b )k ∼ Γ(U = Γ(U = B[[T ]]k . U

U

So we have

b 0, O b0 ) ∼ C ⊗B{T } B[[T ]] ∼ = Γ(U = B[[T ]]k . U

In particular, C ⊗B{T } B[[T ]] is finite and etale over B[[T ]]. This proves our assertion.

7.8

Finiteness of Rf!

([SGA 4] XIV 1, XVI 2, [SGA 4 21 ] Arcata IV 6, V 3.) The main result of this section is the following. Theorem 7.8.1. Let A be a torsion noetherian ring, S a scheme, and f : X → Y an S-compactifiable morphism between noetherian schemes. For any constructible sheaf of A-modules F on X, Rq f! F are constructible for all q. Proof. We only treat the case where nA = 0 for some integer n invertible in S. This is enough for our application. The problem is local on Y . We may assume that Y is affine. Let S = {Z|Z ⊂ X, Z is closed, Rq (f |Z )! (F |Z ) is not constructible for some q}.

If S is not empty, then there exists a minimal member Z in S . Put the reduced closed subscheme structure on Z. Let U be an affine open subset of Z. By 7.4.4 (iii), we have an exact sequence · · · → Rq (f |U )! (F |U ) → Rq (f |Z )! (F |Z ) → Rq (f |Z−U )! (F |Z−U ) → · · · .

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By the minimality of Z, Rq (f |Z−U )! (F |Z−U ) are constructible for all q. If we can prove Rq (f |U )! (F |U ) are constructible for all q, then Rq (f |Z )! (F |Z ) are constructible, which contradicts Z ∈ S . Thus S is empty and Rq f! F are constructible. Therefore it suffices to treat the case where X and Y are affine. Then f : X → Y can be factorized as the composite of a closed immersion i : X → AnY and the projection π : AnY → Y . We have R q f! F ∼ = Rq π! i∗ F .

Note that i∗ F is constructible. So it suffices to consider the case where f = π. Factorizing π as the composites of projections AnY → An−1 → · · · → A1Y → Y Y and applying 7.4.4 (ii), we are reduced to the case where f is the projection π : A1Y → Y . Let F be a constructible sheaf of A-modules on A1Y . By 5.8.5, we can find a resolution · · · → F −1 → F 0 → 0 of F such that each F i is of the form pi! A for some affine etale morphism pi : Ui → A1Y . By 7.4.6, we have a biregular spectral sequence E1pq = Rq π! F p ⇒ Rp+q π! F .

To prove that Rq π! F are constructible, it suffices to prove that Rq π! F i are constructible. We have Rq π! F i = Rq (πpi )! A. On the other hand, we have R(πpi )! A ∼ = A ⊗L Z/n R(πpi )! Z/n. We are thus reduced to prove that Rq f! Z/n are constructible for f = πp, where p : U → A1Y is an affine etale morphism. By 5.8.3, it suffices to show that for any irreducible closed subset Z of Y , there exists a nonempty open subset V of Z such that (Rq f! Z/n)|V are locally constant with finite stalks. Put the reduced closed subscheme structure on Z. By the proper base change theorem 7.4.4 (i), we have (Rq f! Z/n)|Z ∼ = Rq fZ! Z/n,

(Rq f! Z/n)|V ∼ = Rq fV ! Z/n,

where fZ : f −1 (Z) → Z and fV : f −1 (V ) → V are the base changes of f . Replacing Y by Z, we may assume that Y is an integral scheme and we

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need to prove that there exists a nonempty open subset V of Y such that (Rq f! Z/n)|V are locally constant with finite stalks. Let K be the function field of Y . Then UK = U ×Y Spec K is a smooth curve over K. We can find a compactification UK ,→ U K → Spec K of UK such that U K is a smooth projective curve over K, and U K − UK is finite over K. By 1.10.9 and 1.10.10, there exists an open subset W of Y and a compactification g

UW = f −1 (W ) ,→ U W → W of fW : f −1 (W ) → W inducing the above compactification of UK such that g|U W −UW is finite and g is smooth projective and pure of relative dimension 1. We have (g|U W −UW )∗ Z/n if q = 0, Rq (g|U W −UW )! Z/n = 0 if q 6= 0. By 5.8.11 (i), (g|U W −UW )∗ Z/n is constructible. By 7.4.4 (iii), we have an exact sequence · · · → Rq−1 (g|U W −UW )! Z/n → Rq fW ! Z/n → Rq g∗ Z/n → · · · . If we can prove that Rq g∗ Z/n are constructible, then Rq fW ! Z/n are constructible, that is, (Rq f! Z/n)|W are constructible. So there exists an open subset V of W such that (Rq f! Z/n)|V are locally constant with finite stalks. It remains to show that Rq g∗ Z/n are constructible. First note that the stalks of Rq g∗ Z/n are finite. Indeed, for any t ∈ W , we have (Rq g∗ Z/n)t¯ ∼ = H q (U W ⊗OW k(t), Z/n) by the proper base change theorem 7.3.1. The groups H q (U W ⊗OW k(t), Z/n) are finite by 7.2.9 (ii). By 7.8.2 below and 5.8.9, Rq g∗ Z/n are locally constant, and hence constructible. Lemma 7.8.2. Let f : X → Y be a smooth proper morphism, and let F be a locally constant sheaf of Z/n-modules on X for some n invertible on Y . Then for all points s, t ∈ Y such that s ∈ {t}, the specialization homomorphisms (Rq f∗ F )s¯ → (Rq f∗ F )t¯ are isomorphisms for all q.

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Proof. Let Yes¯ be the strict localization of Y at s¯. Fix notation by the following commutative diagram ˜

˜i j Xt¯ = X ×Y t¯ → X ×Y Yes¯ ← Xs¯ = X ×Y s¯ ft¯ ↓ f˜ ↓ ↓ fs¯ j i e ¯ t→ Ys¯ ← s¯,

f= F | where vertical arrows are base changes of f . Let F es¯ . For each X×Y Y q, we have a commutative diagram of canonical morphisms f) ← H q (X ×Y Yes¯, R˜j∗˜j ∗ F f) → H q (Xs¯, ˜i∗ R˜j∗ ˜j ∗ F f) H q (Xt¯, ˜j ∗ F k f) ← H (Xt¯, ˜j ∗ F q

↑ f) H (X ×Y Yes¯, F q

By 7.7.5, we have an isomorphism ∼ = ˜∗ ˜ ˜∗ f f→ ˜i∗ F i Rj∗ j F .

→

↑ f). H (Xs¯, ˜i∗ F q

So the rightmost vertical arrow is an isomorphism. By the proper base change theorem 7.3.3, the two horizontal arrows in the square on the right of the diagram are isomorphisms. The canonical morphism f) → H q (Xt¯, ˜j ∗ F f) H q (X ×Y Yes¯, R˜j∗ ˜j ∗ F is also an isomorphism. It follows that all arrows in the above diagram are isomorphisms. By 5.9.5, we have f), (Rq f∗ F )s¯ ∼ = H q (X ×Y Yes¯, F and by the proper base change theorem 7.3.1, we have f). (Rq f∗ F )t¯ ∼ = H q (Xt¯, ˜j ∗ F

Through these isomorphisms, the specialization homomorphism (Rq f∗ F )s¯ → (Rq f∗ F )t¯

is identified with the canonical morphism f) → H q (Xt¯, ˜j ∗ F f) H q (X ×Y Yes¯, F

in the above diagram. So the specialization homomorphism is an isomorphism. Corollary 7.8.3. Let f : X → Y be a smooth proper morphism between noetherian schemes, and let F be a constructible locally constant sheaf of A-modules on X for some noetherian ring A such that nA = 0 for some integer n invertible on Y . Then Rq f∗ F are locally constant for all q. Proof.

Apply 7.8.1, 7.8.2 and 5.8.9.

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Chapter 8

Duality

8.1

Extensions of Henselian Discrete Valuation Rings

Let R be a henselian discrete valuation ring, K its fraction field, m its maximal ideal, and k = R/m its residue field. Denote by v : K ∗ → Z the valuation on K defined by R. Any element π in R with valuation 1 is called a uniformizer. It is a generator of m. Elements with valuation 0 are units of R. Every element in R can be uniquely written as uπ n for some unit u and some nonnegative integer n. So R is a unique factorization domain. In particular, R is normal. Let K 0 be a finite separable extension of K. Then the integral closure 0 R of R in K 0 is finite over R. It is also a henselian discrete valuation ring. Denote by m0 , k 0 , v 0 the maximal ideal, the residue field, and the valuation of R0 . Define the ramification index of the extension K 0 /K to be the integer e such that mR0 = m0e . We have e = v 0 (π) for any uniformizer π of R. Define the degree of inertia of the extension K 0 /K to be f = [k 0 : k]. Proposition 8.1.1. Notation as above. We have (i) ef = [K 0 : K]. (ii) For any a0 ∈ K 0∗ , we have v(NK 0 /K (a0 )) = f v 0 (a0 ). Proof. (i) Using 1.1.3 (ii) and 1.2.9, one can show that R0 is free of finite rank as an R-module. We have R 0 ⊗R K ∼ = K 0. So the rank of R0 as an R-module is [K 0 : K]. We have R0 ⊗R R/m ∼ = R0 /m0e . 411

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The rank of R0 /m0e as an R/m-module is equal to e X

rank(m0i−1 /m0i ) =

i=1

e X

rank(R0 /m0 ) = ef.

i=1

0

So we have [K : K] = ef . (ii) Both sides of the equality define homomorphisms of groups from K 0∗ to Z. If a0 is a unit in R0 , then NK 0 /K (a) is a unit in R, and both sides are zero. Let π be a uniformizer of K. We have v(NK 0 /K (π)) = v(π [K

0

:K]

) = [K 0 : K] = ef = f v 0 (π).

Let π 0 be a uniformizer K 0 . We have π = u0 π 0e for some unit u0 . The equality holds for a0 = u0 and for a0 = π. So it holds for a0 = π 0e and hence for a0 = π 0 . As K 0∗ is generated by π 0 and units in R0 , the equality holds for any a0 ∈ K 0∗ . We say that the finite separable extension K 0 /K is unramified if e = 1 and k 0 is separable over k. This is equivalent to saying that Spec R0 → Spec R is unramified. Note that Spec R0 → Spec R is always flat. So K 0 /K is unramified if and only if Spec R0 → Spec R is etale. We say K 0 /K is totally ramified if e = [K 0 : K]. This is equivalent to saying that f = 1, that is, k 0 = k. Let p be the characteristic of k. We say that K 0 /K is tamely ramified if e is relatively prime to p, and k 0 is separable over k. Lemma 8.1.2. Suppose R is a henselian discrete valuation ring with fraction field K and residue field k. (i) Let K 0 /K be a finite unramified extension, K 00 /K a finite separable extension, R0 (resp. R00 ) the integral closure of R in K 0 (resp. K 00 ), and k 0 (resp. k 00 ) the residue field of R0 (resp. R00 ). Then the canonical map HomR (R0 , R00 ) → Homk (k 0 , k 00 ) is bijective. (ii) For any finite separable extension k 0 of k, there exists a finite unramified extension K 0 /K such that the residue field of the integral closure R0 of R in K 0 is isomorphic to k 0 . Proof. (i) The set HomR (R0 , R00 ) can be identified with the set of R-morphisms HomSpec R (Spec R00 , Spec R0 ). Identifying an R-morphism with its graph, the set HomSpec R (Spec R00 , Spec R0 ) can be identified with the set of sections of the projection Spec (R0 ⊗R R00 ) → Spec R00 . Since K 0 /K is unramified, this projection is etale. Since R00 is henselian, by 2.8.3 (vii) and

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2.3.10 (i), the set of sections of the projection Spec (R0 ⊗R R00 ) → Spec R00 is in one-to-one correspondence with the set of sections of the projection Spec (k 0 ⊗k k 00 ) → Spec k 00 . The set of sections of the projection Spec (k 0 ⊗k k 00 ) → Spec k 00 can be identified with the set Homk (k 0 , k 00 ). Our assertion follows. (ii) We have k 0 ∼ = k[t]/(f0 (t)) for some monic irreducible polynomial f0 (t) ∈ k[t] such that f0 (t) and f00 (t) are relatively prime. Let f (t) ∈ R[t] be a monic polynomial lifting f0 (t). Applying Nakayama’s lemma to the R-module R[t]/(f (t)), one can show that the ideal generated by f (t) and f 0 (t) is R[t]. By 2.3.3, R[t]/(f (t)) is etale over R. Since f0 (t) is irreducible, f (t) is irreducible as a polynomial in R[t]. By the Gauss lemma, f (t) is irreducible as a polynomial in K[t]. Let K 0 = K[t]/(f (t)), let R0 be the integral closure of R in K, and let k(R0 ) be the residue field of R0 . Then K 0 is a field finite and separable over K. The R-algebra homomorphism R[t]/(f (t)) → R0 induces a k-homomorphism k[t]/(f0 (t)) → k(R0 ). It follows that [k[t]/(f0 (t)) : k] ≤ [k(R0 ) : k]. But we have [K 0 : K] = deg(f ) = [k[t]/(f0 (t)) : k]. So we have [K 0 : K] ≤ [k(R0 ) : k].

Combined with 8.1.1 (i), we see that K 0 /K must be an unramified extension and [K 0 : K] = [k(R0 ) : k]. It has the required property. Let R be a henselian discrete valuation ring with fraction field K and residue field k. Fix a separable closure K of K. By 8.1.2, for any finite separable extension k 0 /k, there exists a finite unramified extension K 0 of K contained in K such that the residue field of the integral closure R0 of R in K 0 is k-isomorphic to k 0 . Let e = lim K 0 , K −0→ K /K

0

e = lim R0 , R −0→ K /K

where K goes over the set of all finite unramified extensions of K contained in K. Then the valuation v : K ∗ → Z can be uniquely extended to a

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e ∗ → Z, R e is the corresponding discrete valuation ring, and valuation v : K ˜ e e is an the residue field k of R is a separable closure of k. Moreover, K (infinite) galois extension of K, and we have a canonical isomorphism ∼ Gal(k/k). ˜ e Gal(K/K) =

e the inertia subgroup of Gal(K/K). It is the kernel We call I = Gal(K/K) of the canonical epimorphism ˜ Gal(K/K) → Gal(k/k). Suppose that U → Spec R is a separated etale morphism of finite type such that U is connected. Then by 2.9.2, U is an open subscheme of Spec R0 , where R0 is the integral closure of R in the function field of U . As Spec R0 consists of only two points, if U → Spec R is surjective, then the open immersion U ,→ Spec R0 is also surjective, and hence U ∼ = Spec R0 . Suppose that A is a local essentially etale R-algebra. The above discussion shows that the fraction field K(A) of A is a finite unramified extension of K and e = lim A is the integral closure of R in K(A). So R R0 is the strict −→K 0 /K henselization of R with respect to the separable closure k˜ of k. Proposition 8.1.3. Suppose that R is a strictly henselian discrete valuation ring with fraction field K. Let π be a uniformizer of K, and let p be the characteristic of the residue field of R. (i) For any positive integer n relatively prime to p, K[t]/(tn − π) is a finite tamely ramified extension of K. Any finite tamely ramified extension of K is isomorphic to K[t]/(tn − π) for some positive integer n relatively prime to p. (ii) Let K 0 /K be a finite separable extension. There exists an extension K 00 of K contained in K 0 such that K 00 /K is tamely ramified and [K 0 : K 00 ] is a power of p. Proof. (i) By the Eisenstein criterion, tn −π is an irreducible polynomial in K[t]. So K[t]/(tn −π) is a field. One can verify that the canonical homomorphism R[t]/(tn − π) → K[t]/(tn − π)

is injective. Hence R[t]/(tn − π) is an integral domain. It is finite over R. If m0 is a maximal ideal of R[t]/(tn − π), then m0 lies above the maximal ideal of R. Hence π ∈ m0 . Since tn = π in R[t]/(tn − π), we have t ∈ m0 . On the other hand, we have R[t]/(tn − π, t) ∼ = R[t]/(π, t) ∼ = R/πR.

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Hence R[t]/(tn − π) is a local integral domain and its maximal ideal is generated by t. This implies that R[t]/(tn − π) is a discrete valuation ring and hence integrally closed. It is the integral closure of R in K[t]/(tn − π). It has the same residue field as R, and the ramification index is n. So K[t]/(tn − π) is a tamely ramified extension of K. Suppose that K 0 /K is a finite tamely ramified extension. Since the residue field of R is separably closed, K 0 /K is a totally ramified extension. Let n = [K 0 : K], let R0 be the integral closure of R in K 0 , and let π 0 be a uniformizer of R0 . The valuation of π in K 0 is n. We have π = uπ 0n for some unit u in R0 . Since K 0 /K is tamely ramified, n is relatively prime to p. Let m0 be the maximal ideal of R0 and let k 0 = R0 /m0 . Then k 0 is isomorphic to the residue field of R and hence is separably closed. So tn − u mod m0 is a product of distinct linear polynomials in k 0 [t]. By 2.8.3 (v), tn − u is a product of linear polynomials in R0 [t]. So u has an n-th root √ √ n u in R0 . Replacing π 0 by π 0 n u, we may assume π = π 0n . Since tn − π is an irreducible polynomial in K[t], it is the minimal polynomial of π 0 . We thus have [K[π 0 ] : K] = n. Hence K 0 = K[π 0 ] ∼ = K[t]/(tn − π). (ii) Let R0 be the integral closure of R in K 0 , let k and k 0 be the residue fields of R and R0 , respectively, let e be the ramification index and let f be the degree of inertia. Since k is separably closed, k 0 /k is a purely inseparable extension. Hence f = [k 0 : k] is a power of p. Write e = e0 pi for some nonnegative integer i and some integer e0 relatively prime to p. Let π 0 be a uniformizer of R0 . We have π = uπ 0e for some unit u in R0 . As k 0 is separably closed, the same argument as the proof of (i) shows that √ √ √ i 0 0 0 e0 u exists in R0 . We have π = ( e uπ 0p )e . So e π exists in R0 . Let √ 0 K 00 = K[ e π]. Then K 00 /K is tamely ramified. We have [K 0 : K 00 ] =

[K 0 : K] ef = 0 = pi f. e0 e

So [K 0 : K 00 ] is a power of p.

Let R be a strictly henselian discrete valuation ring, K its fraction field, K a separable closure of K, π a uniformizer of R, and p the characteristic of the residue field of R. For each positive integer n relatively prime to p, √ choose an n-th root n π of π in K. Let [ √ Kt = K[ n π]. (n,p)=1

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Then by 8.1.3, P = Gal(K/Kt ) is a pro-p-group. We call P the wild inertia subgroup of I. Let I = Gal(K/K). The canonical homomorphism √ σ( n π) √ I → lim µn (K), σ 7→ n ←− π (n,p)=1 induces an isomorphism I/P ∼ µn (K), = lim ←− (n,p)=1

where µn (K) = {ζ ∈ K|ζ n = 1} is the group of n-th roots of unity in K. Note that by 2.8.3 (v), µn (K) is isomorphic to the group µn (k) = {ζ ∈ k|ζ n = 1} of n-th roots of unity in k. Proposition 8.1.4. Let R be a strictly henselian discrete valuation ring with fraction field K and residue field k, let K be a separable closure of K, and let I = Gal(K/K). For any torsion I-module M with torsion relatively prime to p = char k, we have canonical isomorphisms  I if q = 0, M H q (I, M ) ∼ = MI (−1) if q = 1,  0 if q 6= 0, 1

where for any torsion abelian group A with torsion relatively prime to p, we set A(−1) = Hom( lim µn (k), A). ←− (n,p)=1

Proof. Let P be the wild inertia subgroup. Taking P -invariant is an exact functor in the category of P -modules with torsion prime to p, and we have MP ∼ = MP for any module M in this category. Indeed, suppose that 0 → M 0 → M → M 00 → 0

is an exact sequence in this category. For any x00 ∈ M 00P , let x ∈ M be a preimage of x00 in M . Then X 1 gx #(P/Px ) gPx ∈P/Px

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is a preimage of x00 in M P , where Px is the stabilizer of x in P . Here 1 multiplication by #(P/P makes sense since M has torsion relatively prime x) to p so that multiplication by p induces an isomorphism on M . It follows that the sequence 0 → M 0P → M P → M 00P → 0 is exact. The homomorphism M → M,

x 7→

1 #(P/Px )

X

gx

gPx ∈P/Px

induces a homomorphism MP → M P .

One can show it is the inverse of the homomorphism M P → MP induced by idM . So M P ∼ = MP . Therefore the Hochschild–Serre spectral sequence E2uv = H u (I/P, H v (P, M )) ⇒ H u+v (I, M ) degenerates, and we have H q (I, M ) ∼ = H q (I/P, M P ) ∼ = H q (I/P, MP ). By 4.3.9, we have ∼ MI, H 0 (I/P, M P ) ∼ = (M P )I/P = H 1 (I/P, MP ) ∼ = (MP )I/P (−1) ∼ = MI (−1), H q (I/P, M P ) = 0 for q 6= 0, 1.

At the end of this section, we give a proof of 7.1.6, which relies only on 8.1.1. Proof of 7.1.6. Since X is smooth over k, the function field K(X) is separably generated over k. So K(X) can be regarded as a finite separable extension of the rational function field k(t). Since X is proper, the extension k(t) ,→ K(X) defines a finite k-morphism φ : X → P1k ,

and X can be identified with the normalization of P1k in K(X). For any Zariski closed point s ∈ |P1k | in P1k , choose an affine open neighborhood V = Spec A of s in P1k . Then φ−1 (V ) ∼ = Spec B for some finite A-algebra B. For any Zariski closed point x ∈ |X| in X, denote by vx (resp. vs ) the valuation of K(X) (resp. k(t)) corresponding to the point x (resp. s), denote by k(x) (resp. k(s)) the residue field of OX,x (resp. OP1k ,s ), denote

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by OeX,x (resp. OeP1k ,s ) the henselization of OX,x (resp. OP1k ,s ), and denote by K(OeX,x ) (resp. K(OeP1k ,s )) the fraction field of OeX,x (resp. OeP1k ,s ). By 2.8.12, we have Y B ⊗A OeP1k ,s ∼ OeX,x . = x ∈ |X| φ(x) = s

So we have

Y

K(X) ⊗k(t) K(OeP1k ,s ) ∼ = For any g ∈ K(X), we thus have NK(X)/k(t) (g) =

Y

x ∈ |X| φ(x) = s

By 8.1.1 (ii), we have

x ∈ |X| φ(x) = s

NK(OeX,x )/K(Oe 1

P ,s k

X

vs (NK(X)/k(t) (g)) =

K(OeX,x ). ) (g).

[k(x) : k(s)]vx (g).

x ∈ |X| φ(x) = s

It follows that deg(g) = deg =

X

X

x∈|X|

[k(x) : k]vx (g)

x∈|X|

=

vx (g)x

X

X

[k(s) : k][k(x) : k(s)]vx (g)

s∈|P1k | x ∈ |X| φ(x) = s

=

X

[k(s) : k]vs (NK(X)/k(t) (g))

s∈|P1k |

= deg(NK(X)/k(t) (g)). To prove deg(g) = 0, it suffices to show deg(NK(X)/k(t) (g)) = 0. We are thus reduced to the case where X = P1k . In this case, any nonzero g ∈ k(t) can be uniquely written as g=λ

n Y

qiki ,

i=1

where λ ∈ k, qi ∈ k[t] are monic irreducible polynomials, and ki are nonzero integers. Let di be the degrees of the polynomials qi . A Zariski closed point

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s in P1k is either the point ∞, in which case −v∞ (g) is the degree of the polynomial g, or a Zariski closed point in A1k corresponding to the maximal ideal defined by a monic irreducible polynomial q, in which case we have 0 if q 6= qi for all i, vs (g) = ki if q = qi . It follows that deg(g) =

X

[k(s) : k]vs (g)

s∈|P1k |

=

k X

di ki + v∞ (g)

i=1

= 0. This proves our assertion. 8.2

Trace Morphisms

([SGA 4] XVIII 1.1, 2.) In this section, we fix a scheme S and an integer n invertible on S. For any S-scheme X, we denote the sheaf µn,X defined in 7.2.1 by µn for simplicity. For any integer d and any sheaf F of Z/n-modules on X, define ( µ⊗d if d ≥ 0, n Z/n(d) = ⊗(−d) H om µn , Z/n if d ≤ 0, F (d) = F ⊗ Z/n(d).

Let f : X → Y be a smooth S-compactifiable morphism pure of relative dimension d for some integer d. In this section, we define a canonical morphism R2d f! f ∗ F (d) → F , called the trace morphism denoted by TrX/Y or Trf . Suppose that f : X → Y is etale. Then f! is left adjoint to f ∗ . We define Trf : f! f ∗ F → F to be the adjunction morphism. Suppose that X is a smooth irreducible projective curve over an algebraically closed field k. By 7.2.9, we have a canonical isomorphism H 2 (X, µn ) ∼ = Pic(X)/nPic(X). The homomorphism deg : Pic(X) → Z induces an isomorphism Pic(X)/nPic(X) ∼ = Z/n.

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We define TrX/k : H 2 (X, µn ) → Z/n to be the composite deg H 2 (X, µn ) ∼ = Pic(X)/nPic(X) → Z/n.

Suppose that X is a smooth irreducible curve over an algebraically closed field, and let X be its smooth compactification. We have an exact sequence · · · → Hc1 (X−X, µn ) → Hc2 (X, µn ) → H 2 (X, µn ) → Hc2 (X−X, µn ) → · · · . Since X − X is finite, we have Hc1 (X − X, µn ) = Hc2 (X − X, µn ) = 0. So we have an isomorphism Hc2 (X, µn ) ∼ = H 2 (X, µn ). We define TrX/k : Hc2 (X, µ) → Z/n to be the composite of this isomorphism and TrX/k . Suppose that X is a smooth curve over an algebraically closed field. Let X1 , . . . , Xm be its irreducible components. We have M Hc2 (X, µn ) = Hc2 (Xi , µn ) i

Hc2 (X, µn )

We defined TrX/k : → Z/n to be the sum of TrXi /k . Let Y be a smooth curve over an algebraically closed field k and let f : X → Y be an etale morphism. TrX/Y : f! µn → µn induces a homomorphism Hc2 (Y, f! µn ) → Hc2 (Y, µn ). Let SX/Y be the composite Hc2 (X, µn ) ∼ = Hc2 (Y, f! µn ) → Hc2 (Y, µn ).

Lemma 8.2.1. Let Y be a smooth curve over an algebraically closed field k and let f : X → Y be an etale morphism. We have TrY /k ◦ SX/Y = TrX/k . Proof. We may reduce to the case where X and Y are irreducible. Let X and Y be the smooth compactifications of X and Y , respectively, and let f¯ : X → Y be the morphism induced by f . Then f¯ is finite and flat. (The flatness can be proved using 1.1.3 (ii).) So f¯∗ OX is a locally free OY module of finite rank. If V is an open subset of Y such that (f¯∗ OX )|V is a free OY |V -module, then multiplication by a section s ∈ (f¯∗ OX )(V ) defines an endomorphism on the free OY (V )-module (f¯∗ OX )(V ). Its determinant det(s) is a section in OY (V ). We thus get a morphism of sheaves ∗ → OY∗ . det : f¯∗ OX

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Applying this construction after etale base changes, we can define a morphism of etale sheaves ∗ det : f¯∗ OX → OY∗ et . et

Consider the diagram

n ∗ ∗ → f¯∗ OX → 0 f¯∗ µn → f¯∗ OX et et TrX/Y ↓ ↓ det ↓ det n → OY∗ → 0, 0→ µn → OY∗

0→

et

et

where the horizontal lines are exact by Kummer’s theory and the fact that f¯ is finite, and the morphism : f¯∗ µn → µn Tr X/Y

is defined to be the morphism that makes the diagram commute. More explicitly, for any y ∈ Y , choose a separable closure k(y) of k(y). For any x ∈ X ⊗OY k(y), let OeX,¯x (resp. OeY ,¯y ) be the strict henselization of X ¯ (resp. y¯). We have (resp. Y ) at x M ¯ (f∗ µn )y¯ ∼ Γ(Spec Oe , µn ). = Γ(X ⊗O Oe , µn ) ∼ = Y

Y ,¯ y

X,¯ x

x∈X⊗O k(y)

For any (λx ) ∈

Y

L

e , µn ), we have x x∈X⊗O k(y) Γ(Spec OX,¯ Y Y λnx x , TrX/Y ((λx )) = x∈X⊗O k(y) Y

where nx = rankOe OeX,¯x . Let i : X ,→ X and j : Y ,→ Y be the open Y ,y ¯ immersions. One can show the diagram j! f! µn ∼ = f¯∗ i! µn j! (TrX/Y ) & ↓ j! µn

f¯∗ (Tri )

→

Trj

→

f¯∗ µn ↓ TrX/Y µn

commutes by showing it commutes on stalks. Applying H 2 (Y , −) to this diagram, we get a commutative diagram H 2 (Y , j! f! µn ) ∼ = H 2 (Y , f¯∗ i! µn ) → H 2 (Y , f¯∗ µn ) &

↓ ↓ H 2 (Y , j! µn ) → H 2 (Y , µn ).

So we have a commutative diagram

∼ =

Hc2 (X, µn ) → H 2 (X, µn ) SX/Y ↓ ↓ SX/Y ∼ = 2 2 Hc (Y, µn ) → H (Y , µn ),

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where SX/Y is the composite H 2 (X, µn ) ∼ = H 2 (Y , f¯∗ µn )

H 2 (Y ,TrX/Y )

→

Consider the diagram ∼ =

Hc2 (X, µn ) TrX/k SX/Y

↓

Hc2 (Y, µn )

TrY /k

& %

→ Z/n ∼ =

→

H 2 (Y , µn ).

H 2 (X, µn ) . TrX/k - TrY /k

↓ SX/Y H 2 (Y , µn ).

We need to prove that the triangle on the left commutes. We have seen that the outer loop commutes. So it suffices to show that the triangle on the right commutes. By the definition of SX/Y and Kummer’s theory, we have a commutative diagram · · · → Pic(X) → H 2 (X, µn ) → 0 det ↓ ↓ SX/Y · · · → Pic(Y ) → H 2 (Y , µn ) → 0, where det : Pic(X) → Pic(Y ) coincides with the homomorphism ∗ ) H 1 (Y , f∗ OX et

H 1 (Y ,det)

through the identifications ∗ Pic(X) ∼ ), = H 1 (Y , f∗ OX et

→

H 1 (Y , OY∗ et )

Pic(Y ) ∼ = H 1 (Y , OY∗ et ).

So it suffices to prove that the following diagram commutes: Pic(X) det

↓

Pic(Y )

& deg

Z.

% deg

Let L be an invertible OX -module. We claim that L is defined by a Cartier divisor of the form (si , f¯−1 (Vi )), where si are nonzero elements in the function field K(X) of X, {Vi } is an open covering of Y , and ssji ∈ ∗ ¯−1 OX (f (Vi ) ∩ f¯−1 (Vj )). To see that such a Cartier divisor exists, it suffices to find an open covering {Vi } of Y such that L |f¯−1 (Vi ) is free for each i. Let us prove that for any closed point y in Y , there exists an open neighborhood V of y in Y such that L |f¯−1 (V ) is free. Let W = Spec A be an affine open neighborhood of y in Y . Then f¯−1 (W ) = Spec B

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for some finite A-algebra B. Let L be a B-module such that ∼ L | ¯−1 = L∼ , f

(W )

and let m be the maximal ideal of A corresponding to y. Since L∼ is invertible and f¯−1 (y) ∼ = Spec (B ⊗A A/m) is discrete, we can find an isomorphism of (B ⊗A A/m)-modules ∼ =

B ⊗A A/m → L ⊗A A/m.

Choose t ∈ L so that t ⊗ 1 ∈ L ⊗A A/m is the image of 1 in B ⊗A A/m. Let φ : B → L be the B-module homomorphism mapping 1 to t. By Nakayama’s lemma, the homomorphism φ ⊗ idAm : B ⊗A Am → L ⊗A Am

is surjective. So there exists s ∈ A − m such that the homomorphism φ ⊗ idAs : B ⊗A As → L ⊗A As

is surjective. Let V = D(s). Then φ induces an epimorphism Of¯−1 (V ) → L |f¯−1 (V ) .

It is necessarily an isomorphism since its kernel is torsion free and the rank of the stalk of the kernel at the generic point of f¯−1 (V ) is 0. This proves our assertion. Let (si , f¯−1 (Vi )) be a Cartier divisor as above. det(L ) is defined by the Cartier divisor (det(si ), Vi ). To prove deg(det(L )) = deg(L ), it suffices to show X vy (det(s)) = vx (s) x∈f¯−1 (y)

∗

for any s ∈ K(X) and any closed point y of Y , where vy (resp. vx ) is the valuation of the function field K(Y ) (resp. K(X)) of Y (resp. X) at y (resp. x). Making the base change Spec ObY,y → Y , our assertion follows from 8.1.1 (ii). Given an invertible OX -module L on a scheme X, denote by c1 (L ) the image of L under the canonical homomorphism Pic(X) → H 2 (X, µn )

defined via Kummer’s theory. For any morphism f : X → Y , denote by c1X/Y (L ) the image of c1 (L ) under the canonical homomorphism H 2 (X, µn ) → Γ(Y, R2 f∗ µn ). Lemma 8.2.2. Let f¯ : P1Y → Y be the projection. The morphism Z/n → R2 f¯∗ µn mapping 1 to c1P 1 /Y (OP1Y /Y (1)) is an isomorphism. Y

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Proof. Using the proper base change theorem 7.3.1, we may reduce to the case where Y = Spec k for an algebraically closed field k. We have a canonical isomorphism Pic(P1 )/nPic(P1 ) ∼ = H 2 (P1 , µn ) k

k

k

and deg : Pic(P1k ) → Z induces an isomorphism Pic(P1 )/nPic(P1 ) ∼ = Z/n. k

k

Our assertion follows from the fact that OP1k (1) has degree 1.

We define TrP1Y /Y : R f¯∗ µn → Z/n to be the inverse of the isomorphism in 8.2.2. Let j : A1Y → P1Y be the open immersion and let f : A1Y → Y be the projection. Define TrA1Y /Y : R2 f! µn → Z/n to be the composite 2

R 2 f ! µn ∼ = R2 f¯∗ j! µn

R2 f¯∗ (Trj )

→

R2 f¯∗ µn

TrP1

/Y

Y →

Z/n.

Note that TrA1Y /Y commutes with any base change. Let g : X → Y and h : Y → Z be two smooth and S-compactifiable morphisms pure of relative dimensions d and e, respectively. Then f = hg is smooth, S-compactifiable, and pure of relative dimension d+e. By 7.4.5, we have Rq f! F = 0 (resp. Rq g! F = 0, resp. Rq h! F = 0) for any q > 2(d + e) (resp. q > 2d, resp. q > 2e) and any torsion sheaf F on X (resp. X, resp. Y ). Suppose that we have defined TrX/Y : R2d g! Z/n(d) → Z/n,

They induce morphisms

TrX/Y : Rg! Z/n(d)[2d] → Z/n,

TrY /Z : R2e h! Z/n(e) → Z/n. TrY /Z : Rh! Z/n(e)[2e] → Z/n

in the derived categories D(Y, Z/n) and D(Z, Z/n), respectively. We have ∼ Z/n(d) ⊗L g ∗ Z/n(e). Z/n(d + e) = Z/n

By the projection formula 7.4.7, we have Rg! Z/n(d) ⊗L g ∗ Z/n(e) ∼ = Rg! Z/n(d) ⊗L Z/n

Z/n

Z/n(e).

We define

TrX/Z : Rf! Z/n(d + e)[2(d + e)] → Z/n

to be the composite

Rf! Z/n(d + e)[2(d + e)]

∗ Rh! Rg! Z/n(d) ⊗L Z/n g Z/n(e) [2(d + e)] Rh! Rg! Z/n(d)[2d] ⊗L Z/n Z/n(e)[2e] TrX/Y → Rh! Z/n ⊗L Z/n Z/n(e)[2e] ∼ Rh! Z/n(e)[2e] = ∼ = ∼ =

TrY /Z

→ Z/n.

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We define 2(d+e)

TrX/Z : Rf!

Z/n(d + e) → Z/n

to be the morphism induced by TrX/Z : Rf! Z/n(d + e)[2(d + e)] → Z/n. We denote the above way of defining TrX/Z by TrX/Z = TrY /Z TrX/Y . Next we define TrAdY /Y . This has been done for d = 0, 1. We can factorize the projection AdY → Y as the composite of projections AdY → Ad−1 → · · · → A1Y → Y, Y

where for each i, AiY → Ai−1 is the projection Y

Spec OY [t1 , . . . , ti ] → Spec OY [t1 , . . . , ti−1 ] to the first i − 1 coordinates. We define TrAdY /Y = TrA1Y /Y · · · TrAd /Ad−1 , Y

Y

where for each i, we define TrAi

Y

/Ai−1 Y

= TrA1 A

i−1 i−1 /AY Y

.

If an S-compactifiable morphism f : X → Y can be factorized as g

X → AdY → Y

with g being an etale Y -morphism, then we define TrX/Y = TrAdY /Y Trg . The following lemma shows that TrX/Y is independent of the choice of the factorization. Lemma 8.2.3. Let f : X → Y be an S-compactifiable morphism, let x1 , . . . , xd ∈ Γ(X, OX ) be a sequence such that the Y -morphism g : X → AdY = Spec OY [t1 , . . . , td ]

corresponding to the OY -algebra homomorphism g \ : OY [t1 , . . . , tn ] → f∗ OX ,

ti 7→ xi

is etale, and let T(x1 ,...,xd ) = TrAdY /Y Trg . Then T(x1 ,...,xd ) is independent of the choice of the sequence x1 , . . . , xd such that the corresponding morphism g : X → AnY is etale.

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Proof. By base change, we may assume that Y = Spec k for some algebraically closed field k. If d = 1, then X is a smooth curve over k. By 8.2.1, Tx1 coincides with TrX/k and hence does not depend on the choice of x1 . In general, we may assume that X is irreducible. For any nonempty open subset U of X, we have Hcq (X − U, Z/n(d)) = 0 for all q > 2(d − 1) by 7.4.5. It follows that the canonical homomorphism Hc2d (U, Z/n(d)) → Hc2d (X, Z/n(d))

is an isomorphism. To prove our assertion, we may replace X by its nonempty open subset. First we prove that T(x1 ,...,xd ) is independent of the order of x1 , . . . , xd . In fact, we prove that, more generally, for any matrix (aij ) ∈ GLd (k), we have T(x1 ,...,xd ) = T(Pi ai1 xi ,...,Pi aid xi ) . Let g : X → Adk (resp. g 0 : X → Adk ) be the k-morphism defined by the P P d d sections x1 , . . . , xd (resp. i ai1 xi , . . . , i aid xi ), and let φ : Ak → Ak be the k-isomorphism corresponding to the k-algebra isomorphism X k[t1 , . . . , tn ] → k[t1 , . . . , tn ], tj 7→ aij ti . i

By definition, we have

T(x1 ,...,xd ) = TrAdk /k ◦ R2d π! (Trg ),

T(Pi ai1 xi ,...,Pi aid xi ) = TrAdk /k ◦ R2d π! (Trg0 ), where π : Adk → Spec k is the projection. To prove our assertion, it suffices to show R2d π! (Trg ) = R2d π! (Trg0 ). We have g 0 = φg. So Trg0 is the composite g!0 Z/n ∼ = φ! g! Z/n

φ! (Trg )

→

Trφ

φ! Z/n → Z/n.

We have πφ = π. To prove R2d π! (Trg ) = R2d π! (Trg0 ), it suffices to show that the composite R2d π! Z/n ∼ = R2d π! φ! Z/n

R2d π! (Trφ )

→

R2d π! Z/n

is the identity. Indeed, let Φ be the isomorphism Φ : GLd (k) ×k Adk → GLd (k) ×k Adk ,

X (xij ), (xj ) 7→ (xij ), ( xij xi ) . i

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Fix notation by the following commutative diagram: p2

Φ

GLd (k) ×k Adk → GLd (k) ×k Adk → Adk p1 & p1 ↓ π ↓ GLd (k) → Spec k. The above composite is the stalk at the geometric point of GLd (k) defined by (aij ) of the following composite: R R2d p1! Z/n ∼ = R2d p1! Φ! Z/n

2d

p1! (TrΦ )

→

R2d p1! Z/n.

But R2d p1! Z/n is isomorphic to the inverse image of R2d π! Z/n under the morphism GLd (k) → Spec k, and hence is a constant sheaf. As GLd (k) is connected, the stalk of the last composite at the geometric point defined by (aij ) can be identified with the stalk at the geometric point defined by the identity matrix. Our assertion follows. Let x be a closed point of X and let m be the maximal ideal of OX,x . For any two sequences (x1 , . . . , xd ) and (y1 , . . . , yd ) of Γ(X, OX ) such that the k-morphisms X → Adk are etale, we claim that after replacing X by an open neighborhood of x, there exists a finite chain of sequences m (x1 , . . . , xd ) = (x01 , . . . , x0d ), (x11 , . . . , x1d ), · · · , (xm 1 , . . . , xd ) = (y1 , . . . , yn )

such that the k-morphisms X → Adk defined by these sequences are etale, and for each v ∈ {0, 1, . . . , m − 1}, the sequence (xv1 , . . . , xvd ) differs from (xv+1 , . . . , xv+1 ) by only one element. We may assume that x1 , . . . , xd lie 1 d in m by replacing xi by xi − ai for some ai ∈ k with xi ≡ ai mod m. Then (x1 , . . . , xd ) is a regular system of parameters for the regular local ring OX,x . Indeed, since the morphism X → Adk is etale, dx1 , . . . , dxd form a basis for the free OX,x -module ΩOX,x /k . By 2.1.5, the canonical homomorphism m/m2 → ΩOX,x /k ⊗OX,x OX,x /m

∼ k. It follows that the images of is an isomorphism since OX,x /m = x1 , . . . , xd in m/m2 form a basis for the (OX,x /m)-vector space m/m2 . Similarly, we may assume that y1 , . . . , yn lie in m and form a regular system of parameters for OX,x . Any basis of m/m2 can be changed to another basis by a chain of bases so that each successive pair of bases differ only by one element. So we have a chain of regular system of parameters m (x1 , . . . , xd ) = (x01 , . . . , x0d ), (x11 , . . . , x1d ), · · · , (xm 1 , . . . , xd ) = (y1 , . . . , yn )

of OX,x such that for each v ∈ {0, 1, . . . , m − 1}, the sequence (xv1 , . . . , xvd ) differs from (xv+1 , . . . , xv+1 ) by just one element. Replacing X by an open 1 d

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neighborhood of x, we may assume that elements of these sequences can be extended to sections in Γ(X, OX ), such that the k-morphisms X → Ank defined by them are etale. This proves our claim. By the above discussion, to prove that T(x1 ,...,xd ) is independent of the choice of (x1 , . . . , xd ), it suffices to show that T(x1 ,...,xd ) remains unchanged if we change xd but keep x1 , . . . , xd−1 unchanged. Consider the commutative diagram g

AdY ∼ = A1Ad−1 Y

%↓ h

X → Ad−1 Y &↓ f

Y, where g : X → AdY (resp. h : X → Ad−1 Y ) is the Y -morphism defined by the sections x1 , . . . , xd (resp. x1 , . . . , xd−1 ). By our discussion in the d = 1 case, TrAd /Ad−1 Trg is unchanged if we only change xd . We have Y

Y

T(x1 ,...,xd ) = TrAdY /Y Trg

= TrAd−1 /Y TrAd /Ad−1 Trg . Y

Y

Y

The last expression is independent of xd . So T(x1 ,...,xd ) is independent of the choice of xd . Finally, let f : X → Y be a smooth S-compactifiable morphism pure of relative dimension d. There exists an open covering {Uα } of X such that f |Uα : Uα → Y can be factorized as gα

Uα → AdY → Y, with gα being etale Y -morphisms. Let jα : Uα ,→ X and jαβ : Uα ∩Uβ → X be the open immersions. For any sheaf of Z/n-modules F on X, we have a canonical exact sequence M M jαβ! (F |Uα ∩Uβ ) → jα! (F |Uα ) → F → 0. α

α,β

This follows from the fact that for any sheaf of Z/n-modules G on X, we have an exact sequence M M 0 → Hom(F , G ) → Hom jα! (F |Uα ), G → Hom jαβ! (F |Uα ∩Uβ ), G , α

α,β

etale

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which can be identified with the canonical exact sequence Y Y 0 → Hom(F , G ) → Hom(F |Uα , G |Uα ) → Hom(F |Uα ∩Uβ , G |Uα ∩Uβ ). α

α,β

q

By 7.4.5, we have R f! = 0 for all q > 2d. So R2d f! is right exact. So we have an exact sequence M M R2d (f |Uα ∩Uβ )! (F |Uα ∩Uβ ) → R2d (f |Uα )! (F |Uα ) → R2d f! F → 0. α

α,β

We have defined

TrUα /Y : R2d (f |Uα )! Z/n(d) → Z/n. By 8.2.3, TrUα /Y and TrUβ /Y both induce TrUα ∩Uβ /Y . Using the above exact sequence, we can define TrX/Y : R2d f! Z/n(d) → Z/n so that it induces TrUα /Y for each α. The definition of TrX/Y is independent of the choice of the open covering {Uα } of X. Since Ri f! = 0 for i > 2d, TrX/Y induces a morphism Rf! Z/n(d)[2d] → Z/n in the derived category D(Y, Z/n). For any K ∈ ob D(Y, Z/n), we have Rf! Z/n(d)[2d] ⊗L K ∼ = Rf! (f ∗ K(d)[2d]) Z/n

by the projection formula 7.4.7. So TrX/Y induces a morphism Rf! f ∗ K(d)[2d] → K, which we also denote by TrX/Y or Trf . It is functorial in K. Proposition 8.2.4. Let S be a scheme and let n be an integer invertible on S. (i) Consider a Cartesian diagram g0

X ×Y Y 0 → X f0 ↓ ↓f g 0 Y → Y.

Suppose that f is a smooth S-compactifiable morphism pure of relative dimension d, and Y 0 is quasi-compact and quasi-separated. For any K ∈ ob D(Y, Z/n), the following diagram commutes: g ∗ Rf! f ∗ K(d)[2d] ∼ = Rf 0 g 0∗ f ∗ K(d)[2d] ∼ = Rf 0 f 0∗ g ∗ K(d)[2d] !

g∗ (Trf )

&

!

g ∗ K.

. Trf 0

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(ii) Let g : X → Y and h : Y → Z be two smooth S-compactifiable morphisms pure of relative dimensions d and e, respectively and let f = hg. For any K ∈ ob D(Z, Z/n), the following diagram commutes: Rh! Rg! g ∗ h∗ K(d + e)[2(d + e)] ∼ = ↓

Rh! (Trg )

→

Trf

Rf! f ∗ K(d + e)[2(d + e)]

→

Rh! h∗ K(e)[2e] ↓ Trh K.

(iii) Consider a Cartesian diagram p

X ×Z Y → X q ↓ ↓f g Y → Z. Let h = f p = gq. Suppose that f and g are smooth S-compactifiable morphisms pure of relative dimensions d and e, respectively. For any K, L ∈ ob D− (Z, Z/n), the following diagram commutes: ∗ ∗ L ∼ Rf! f ∗ K(d)[2d] ⊗L Z/n Rg! g L(e)[2e] = Rh! h (K ⊗Z/n L)(d + e)[2(d + e)] Trf ⊗L Trg & . Trh L K ⊗Z/n L,

where the top line is given by the K¨ unneth formula. (iv) Suppose that S = Spec k for an algebraically closed field k, X and Y are proper smooth schemes over k pure of dimensions d and e, respectively. Then for any s ∈ H 2d (X, Z/n(d)) and t ∈ H 2e (Y, Z/n(e)), we have TrX×k Y /k (p∗ s ∪ q ∗ t) = TrX/k (s)TrY /k (t), where p : X ×k Y → X and q : X ×k Y → Y are projections. Proof. (i) follows from the definition of Tr. (iv) follows from (iii) and the fact that the isomorphism defined by the K¨ unneth formula can be defined via the cup product. (Confer the proof of 7.4.11.) (ii) We can reduce to the case where we have a commutative diagram a

X g

→ AdY & π00 ↓ Y

b0

→ Ad+e Z ↓ π0 b

→ AeZ h& ↓π Z

etale

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such that the square in the diagram is Cartesian, a, b, b0 are etale, and π, π 0 , π 00 are the projections. Consider the diagram Rπ! b! Rπ!00 a! Z/n(d + e)[2(d + e)] ∼ = Rπ! Rπ!0 b0! a! Z/n(d + e)[2(d + e)] Tra ↓ ↓ Tra Rπ! b! Rπ!00 Z/n(d + e)[2(d + e)] ∼ = Rπ! Rπ!0 b0! Z/n(d + e)[2(d + e)] ↓ Trb0 Trπ00 ↓ (1) Rπ! Rπ!0 Z/n(d + e)[2(d + e)] ↓ Trπ0 Tr

Tr

Rπ! b! Z/n(e)[2e] →b Rπ! Z/n(e)[2e] →π Z/n. To prove that the square (1) commutes, it suffices to prove that the diagram b! Rπ!00 Z/n(d)[2d)] ∼ = Rπ!0 b0! Z/n(d)[2d] ↓ Trb0 Trπ00 ↓ (2) Rπ!0 Z/n(d)[2d] ↓ Trπ0 Tr

b! Z/n →b Z/n commutes if b, b0 , π 0 , π 00 come from the following Cartesian diagram b0

AdY → AdW π 00 ↓ ↓ π0 b

Y →W in which b is etale. Let us check that (2) gives rise to a commutative diagram on stalks at each geometric point s → W of W . Making the base change s → W , we are reduced to the case where W is the spectrum of a separably closed field. We are then reduced to the case where b is the identity morphism. In this case the diagram (2) trivially commutes. (iii) follows from the commutativity of the following diagram: ∗ Rf! f ∗ K(d)[2d] ⊗L Z/n Rg! g L(e)[2e] ok ∗ Rg! (g ∗ Rf! f ∗ K(d)[2d] ⊗L Z/n g L(e)[2e]) ok ∗ Rg! (Rq! p∗ f ∗ K(d)[2d] ⊗L Z/n g L(e)[2e]) ok ∗ Rg! (Rq! q ∗ g ∗ K(d)[2d] ⊗L Z/n g L(e)[2e]) ok

Trf ⊗L id

→

Rg! (g ∗ Trf ⊗L id)

→

∗ K ⊗L Z/n Rg! g L(e)[2e] ok ∗ Rg! (g ∗ K ⊗L Z/n g L(e)[2e])

k Rg! (Trq ⊗L id)

→

∗ Rg! (g ∗ K ⊗L Z/n g L(e)[2e]) ok

Rg! Rq! q ∗ g ∗ (K(d)[2d] ⊗L Z/n L(e)[2e]) ok

Rg! (Trq )

→

Rg! g ∗ (K ⊗L Z/n L(e)[2e]) ↓ Trg

Rh! h∗ (K ⊗L Z/n L)(d + e)[2(d + e)]

Trh

K ⊗L Z/n L.

→

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8.3

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Duality for Curves

([SGA 4 21 ] Dualit´e.) In this section, we fix an integer n invertible in all schemes that we consider, and we denote ExtZ/n (−, −) and E xtZ/n (−, −) by Ext(−, −) and E xt(−, −), respectively. Let X be a scheme over an algebraically closed field k. For any sheaves of Z/n-modules F and G on X, we have canonical homomorphisms Exti (F , G ) ∼ = HomD(X,Z/n) (F , G [i]) → HomD(Z/n) RΓc (X, F ), RΓc (X, G [i]) → Hom Hcj (X, F ), Hci+j (X, G ) .

We thus get a pairing

Hcj (X, F ) × Exti (F , G ) → Hci+j (X, G ).

Suppose that X is smooth pure of dimension d. We have a pairing Hcr (X, F ) × Ext2d−r (F , Z/n(d)) → Hc2d (X, Z/n(d)).

Taking its composite with

TrX/k : Hc2d (X, Z/n(d)) → Z/n,

we get a pairing

Hcr (X, F ) × Ext2d−r (F , Z/n(d)) → Z/n.

In 8.5.3, we prove that this last pairing is perfect. In this section, we prove this result in the case where X is a smooth curve over k and F is constructible. Theorem 8.3.1. Let X be a smooth curve over an algebraically closed field k, and let F be a constructible sheaf of Z/n-modules on X. The canonical pairing Hcr (X, F ) × Ext2−r (F , µn ) → Z/n

is perfect for each r.

A direct consequence of 8.3.1 is the following: Corollary 8.3.2. Let X be a smooth curve over an algebraically closed field k, let F be a locally constant constructible sheaf of Z/n-modules on X, and let F ∨ = H om(F , Z/n). Then we have a perfect pairing for each r.

Hcr (X, F ) × H 2−r (X, F ∨ (1)) → Z/n

etale

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Proof.

etale

433

It suffices to show Ext2−r (F , µn ) ∼ = H 2−r (X, H om(F , µn )).

We have a biregular spectral sequence E2pq = H p (X, E xtq (F , µn )) ⇒ Extp+q (F , µn ). By 8.3.3 below, we have E xtq (F , µn ) = 0 for all q ≥ 1. The spectral sequence degenerates. Our assertion follows. Lemma 8.3.3. Let X be a noetherian scheme, let A be a ring such that A is an injective A-module, and let F be a locally constant sheaf of A-modules on X. Then E xtqA (F , A) = 0 for all q ≥ 1. Remark 8.3.4. Note that Z/n is an injective Z/n-module. To prove this, it suffices to show that any homomorphism from an ideal I of Z/n to Z/n can be extended to an endomorphism of Z/n. This follows from the fact that I is generated by an element ¯i for some integer i dividing n. Similarly, one can show that if A = R/I for a discrete valuation ring R and a nonzero ideal I of R, then A is an injective A-module. Proof of 8.3.3. The problem is local with respect to the etale topology. We may assume that F is a constant constructible sheaf associated to some A-module M . Let · · · → L1 → L0 → 0 be a resolution of M by free A-modules. Denote the constant sheaf on X associated to Li also by Li . Then E xtqA (Li , G ) = 0 for any q ≥ 1 and any sheaf of A-modules G on X. It follows that E xtqA (F , A) is the q-th cohomology sheaf of the complex 0 → H omA (L0 , A) → H omA (L1 , A) → · · · for each q. The q-th cohomology sheaf of this complex is the constant sheaf associated to the q-th cohomology of the complex of A-modules 0 → HomA (L0 , A) → HomA (L1 , A) → · · · . Since A is an injective A-module, for any q ≥ 1, the q-th cohomology of this complex is 0. Our assertion follows.

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A scheme is called a trait (resp. strictly local trait) if it is isomorphic to the spectrum of a henselian discrete valuation ring (resp. strictly henselian discrete valuation ring). Lemma 8.3.5. Let S be a strictly local trait, s its closed point, η its generic point, I = Gal(¯ η /η), i : s → S and j : η → S the immersions. For any sheaf of Z/n-modules F on η, we have canonical isomorphisms  if q = 0,  (Fη¯ )I q (R j∗ F )s = (Fη¯ )I (−1) if q = 1,  0 if q 6= 0, 1. Proof.

Since S is strictly local, we have (Rj∗ F )s ∼ = RΓ(S, Rj∗ F ) ∼ = RΓ(η, F ).

It follows that (Rq j∗ F )s = H q (I, Fη¯ ). We then apply 8.1.4. Lemma 8.3.6. Let X be a noetherian regular scheme pure of dimension 1, x a closed point of X, and i : Spec k(x) → X the closed immersion. For any constant sheaf of Z/n-module M on X, we have canonical isomorphisms M (−1) if q = 2, q ! ∼ R iM = 0 if q 6= 2. Proof.

ex¯ be the strict localization of X at x Let X ¯, and let e ex¯ ˜ j : X − {x} → X, j : Xx¯ ×X (X − {x}) → X

ex¯ is a strictly local trait, and X ex¯ ×X be the open immersions. Note that X e (X − {x}) is the generic point of Xx¯ . By 8.3.5, we have  if q = 0, M (Rq j∗ j ∗ M )x¯ ∼ = (Rq ˜j∗ ˜j ∗ M )x¯ = M (−1) if q = 1,  0 if q 6= 0, 1. Note that the canonical morphism M → j∗ j ∗ M

is an isomorphism. This is clear when restricted to X −x. At x, this follows from the above formula for (j∗ j ∗ M )x¯ . We have an exact sequence 0 → i∗ i! M → M → j∗ j ∗ M → i∗ R1 i! M → 0

and isomorphisms

i∗ Rq+1 i! M ∼ = Rq j∗ j ∗ M for all q ≥ 1. Our assertion follows.

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Lemma 8.3.7. Let X be a smooth curve over an algebraically closed field k, let π : X 0 → X be an etale morphism and let F 0 be a constructible sheaf of Z/n-modules on X 0 . For each r, the pairing Hcr (X 0 , F 0 ) × Ext2−r (F 0 , µn ) → Z/n

is perfect if and only if the pairing is perfect. Proof.

Hcr (X, π! F 0 ) × Ext2−r (π! F 0 , µn ) → Z/n Since π is etale, we have H r (X, π! F 0 ) ∼ = H r (X 0 , F 0 ), c

c

Ext2−r (F 0 , µn ) ∼ = HomD(X 0 ,Z/n) (F 0 , π ∗ µn [2 − r]) ∼ = HomD(X 0 ,Z/n) (π! F 0 , µn [2 − r]) ∼ = Ext2−r (π! F 0 , µn ).

Our assertion follows from the commutativity of the following diagram: Hcr (X 0 , F 0 ) × Ext2−r (F 0 , µn ) → Hc2 (X 0 , µn ) ok ↓ ok TrX 0 /k & Hcr (X, π! F 0 ) × Ext2−r (π! F 0 , π! µn ) → Hc2 (X, π! µn ) Z/n. k ↓ Trπ ↓ Trπ TrX/k % Hcr (X, π! F 0 ) × Ext2−r (π! F 0 , µn ) → Hc2 (X, µn ) Lemma 8.3.8. 8.3.1 holds if F is supported on a finite closed subset of X. Proof. Working with each irreducible component of X, we may assume that X is irreducible. If F is supported on a finite closed subset, then F is a direct sum of sheaves supported on a single point. So it suffices to consider the case where F = i∗ M for some finite Z/n-module M and some closed immersion i : {x} → X. By 8.3.7, we may replace X by its smooth compactification. So we may assume that X is projective. We have M if r = 0, r ∼ H (X, i∗ M ) = 0 if r 6= 0, and Ext2−r (i∗ M, µn ) ∼ = HomD(X,Z/n) (i∗ M, µn [2 − r]) ∼ = HomD(Z/n) (M, Ri! µn [2 − r]) ∼ = HomD(Z/n) (M, Z/n[−r]) Hom(M, Z/n) if r = 0, ∼ = 0 if r 6= 0.

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Here we use 8.3.6 and the fact that Z/n is an injective Z/n-module. Since the canonical pairing M × Hom(M, Z/n) → Z/n is perfect, the pairing H 0 (x, M ) × Ext2 (M, Ri! µn ) → H 2 (x, Ri! µn ) is perfect. We claim that each morphism in the composite ∼ =

TrX/k

H 2 (x, Ri! µn ) → H 2 (X, i∗ Ri! µn ) ∼ = Hx2 (X, µn ) → H 2 (X, µn ) → Z/n is an isomorphism. Indeed, since X is smooth projective and irreducible, the homomorphism TrX/k : H 2 (X, µn ) → Z/n is an isomorphism. We have a long exact sequence · · · → H 1 (X−{x}, µn) → Hx2 (X, µn ) → H 2 (X, µn ) → H 2 (X−{x}, µn ) → · · · . By 7.2.13, we have H 2 (X − {x}, µn ) = 0. So the canonical homomorphism Hx2 (X, µn ) → H 2 (X, µn ) is surjective. Since both Hx2 (X, µn ) and H 2 (X, µn ) are free Z/n-modules of rank 1, this homomorphism is an isomorphism. Our claim follows. 8.3.1 then follows from the commutativity of the following diagram: H 0 (X, i∗ M ) × Ext2 (i∗ M, µn ) → H 2 (X, µn ) k ↑ ↑ 2 0 ! 2 H (X, i∗ M ) × Ext (i∗ M, i∗ Ri µn ) → H (X, i∗ Ri! µn ) ok ↑ ok H 0 (x, M ) × Ext2 (M, Ri! µn ) → H 2 (x, Ri! µn )

Tr X/k

&

∼ =

%

Z/n.

Lemma 8.3.9. 8.3.1 holds for F = Z/n. Proof. Working with each irreducible component of X, we may assume that X is irreducible. Let X be a smooth compactification of X, and let j : X ,→ X and i : X − X → X be the immersions. By 8.3.7, it suffices to prove that the pairing H r (X, j! Z/n) × Ext2−r (j! Z/n, µn ) → Z/n

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is perfect for each r. We have an exact sequence 0 → j! Z/n → Z/n → i∗ Z/n → 0. For any Z/n-module M , let M ∨ = Hom(M, Z/n). Since Z/n is an injective Z/n-module, the functor M 7→ M ∨ is exact. Consider the commutative diagram H 0 (X, j! Z/n) → H 0 (X, Z/n) → H 0 (X, i∗ Z/n) → ↓ ↓ ↓ · · · → (Ext2 (j! Z/n, µn ))∨ → (Ext2 (Z/n, µn ))∨ → (Ext2 (i∗ Z/n, µn ))∨ → 0→

H 1 (X, j! Z/n) → H 1 (X, Z/n) → H 1 (X, i∗ Z/n) → ··· ↓ ↓ ↓ → (Ext1 (j! Z/n, µn ))∨ → (Ext1 (Z/n, µn ))∨ → (Ext1 (i∗ Z/n, µn ))∨ → · · ·

→

where the vertical arrows are induced by the pairings, and the horizontal lines are exact. By 8.3.8, the homomorphisms H r (X, i∗ Z/n) → (Ext2−r (i∗ Z/n, µn ))∨ are bijective. To prove our assertion, it suffices to show that the homomorphisms H r (X, Z/n) → (Ext2−r (Z/n, µn ))∨ are bijective. We are thus reduced to the case where X is projective over k. We have seen in the proof of 8.3.2 that we have Ext2−r (Z/n, µn ) ∼ = H 2−r (X, H om(Z/n, µn )) ∼ = H 2−r (X, µn ). So we have Ext2−r (Z/n, µn ) = 0 for r 6= 0, 1, 2. Moreover, combined with 7.2.9, we see that Ext2−r (Z/n, µn ) has the same number of elements as H r (X, Z/n) for each r. Consider the commutative diagram HomD(X,Z/n) (Z/n, Z/n) × HomD(X,Z/n) (Z/n, µn [2]) → HomD(X,Z/n) (Z/n, µn [2]) ∼ ∼ = ↓ = ↓ H 0 (RΓ(X, Z/n)) × Ext2 (Z/n, µn ) → H 0 (RΓ(X, µn [2])) ∼ ∼ = ↓ = ↓ H 0 (X, Z/n) × Ext2 (Z/n, µn ) → H 2 (X, µn ) ∼ = ↓ Tr X/k Z/n,

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where the pairing on the top line is (φ, ψ) 7→ ψφ. For any ψ ∈ HomD(X,Z/n) (Z/n, µn [2]), if ψφ = 0 for all φ ∈ HomD(X,Z/n) (Z/n, Z/n), then we have ψ = 0 by taking φ = id. Since H 0 (X, Z/n) and Ext2 (Z/n, µn ) have the same number of elements, the pairing H 0 (X, Z/n) × Ext2 (Z/n, µn ) → Z/n is perfect. Since H 1 (X, Z/n) and Ext1 (Z/n, µn ) have the same number of elements, to prove that the pairing H 1 (X, Z/n) × Ext1 (Z/n, µn ) → Z/n is perfect, it suffices to show that the homomorphism H 1 (X, Z/n) → (Ext1 (Z/n, µn ))∨

induced by the pairing is a monomorphism. Given α ∈ H 1 (X, Z/n), let π : X 0 → X be the Z/n-torsor defined by α. The image of α under the canonical homomorphism H 1 (X, Z/n) → H 1 (X 0 , Z/n)

is 0 since the base change of the above torsor by X 0 → X has a global section and is hence trivial. It follows that the image of α under the canonical homomorphism H 1 (X, Z/n) → H 1 (X, π∗ π ∗ Z/n) ∼ = H 1 (X, π∗ Z/n) is 0. Since π is surjective, the canonical morphism Z/n → π∗ Z/n is injective. Let F be its cokernel. Consider the commutative diagram H 0 (X,π∗ Z/n) →

H 0 (X,F ) →

↓ 2

H 1 (X,Z/n) →

↓ ∨

2

H 1 (X,π∗ Z/n)

↓ ∨

1

↓ ∨

1

(Ext (π∗ Z/n,µn )) → (Ext (F ,µn )) → (Ext (Z/n,µn )) → (Ext (π∗ Z/n,µn ))∨

We have proved that the pairing H 0 (X, Z/n) × Ext2 (Z/n, µn ) → Z/n is perfect for any smooth projective curve X. In particular, the pairing H 0 (X 0 , Z/n) × Ext2 (Z/n, µn ) → Z/n is perfect. By 8.3.7, the pairing H 0 (X, π∗ Z/n) × Ext2 (π∗ Z/n, µn ) → Z/n

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is perfect. So the first vertical arrow in the above diagram is an isomorphism. Suppose α lies in the kernel of H 1 (X, Z/n) → (Ext1 (Z/n, µn ))∨ . If we can show that H 0 (X, F )) → (Ext2 (F , µn ))∨ is injective, then by diagram chasing and taking into account the fact that α lies in the kernel of H 1 (X, Z/n) → H 1 (X, π∗ Z/n), we can show α = 0. Hence to show that H 1 (X, Z/n) → (Ext1 (Z/n, µn ))∨ is injective, it suffices to show that H 0 (X, F ) → (Ext2 (F , µn ))∨ is injective. Since π is finite and etale, π∗ Z/n is locally constant by 7.8.3. So F is locally constant. It is constructible by 5.8.11. Using 5.8.1 (i), one can show that there exists a finite surjective etale morphism π 0 : X 00 → X such that π 0∗ F is constant. Since every finitely generated Z/n-module can be embedded into a free Z/n-module of finite rank, there exists a free Z/nmodule L of finite rank such that we have a monomorphism π 0∗ F → L. As F → π∗0 π 0∗ F is injective, we have a monomorphism F → π∗0 L. Let G be its cokernel. Consider the commutative diagram H 0 (X, F ) → H 0 (X, π∗0 L) → H 0 (X, G ) → ··· ↓ ↓ ↓ · · · → (Ext2 (F , µn ))∨ → (Ext2 (π∗0 L, µn ))∨ → (Ext2 (G , µn ))∨ → · · · . 0→

By our previous discussion, the pairing H 0 (X 00 , Z/n) × Ext2 (Z/n, µn ) → Z/n is perfect. By 8.3.7, this implies that the homomorphism H 0 (X, π∗0 L) → (Ext2 (π∗0 L, µn ))∨ is bijective. It follows from the above commutative diagram that H 0 (X, F ) → (Ext2 (F , µn ))∨ is injective. Finally we prove that the homomorphism H 2 (X, Z/n) → (Hom(Z/n, µn ))∨ , induced by the paring H 2 (X, Z/n) × Hom(Z/n, µn ) → Z/n,

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is injective. Let α ∈ H 2 (X, Z/n) be an element in the kernel of the above homomorphism. For any morphism φ : Z/n → µn , denote by [φ](α) ∈ H 2 (X, µn ) the image of α under the homomorphism [φ] : H 2 (X, Z/n) → H 2 (X, µn ) induced by φ. Then we have TrX/k ([φ](α)) = 0. As X is smooth projective and irreducible, TrX/k is an isomorphism. So we have [φ](α) = 0. If we take φ to be an isomorphism from Z/n to µn , we get α = 0. Proof of 8.3.1. Since F is constructible, we can find an etale morphism π : X 0 → X together with an epimorphism π! Z/n → F . Let G be the kernel of this epimorphism. It is also constructible. Consider the commutative diagram 0 → Ext0 (F , µn ) → Ext0 (π! Z/n, µn ) → Ext0 (G , µn ) → ↓ (1) ↓ (2) ↓ (3) 0 → Hc2 (X, F )∨ → Hc2 (X, π! Z/n)∨ → Hc2 (X, G )∨ → → Ext1 (F , µn ) → Ext1 (π! Z/n, µn ) → Ext1 (G , µn ) → · · · ↓ (4) ↓ (5) ↓ (6) → Hc1 (X, F )∨ → Hc1 (X, π! Z/n)∨ → Hc1 (X, G )∨ → · · ·

By 8.3.7 and 8.3.9, (2) is bijective. So (1) is injective. This is true for any constructible sheaf F . So (3) is injective. It then follows that (1) is bijective. This is true for any constructible sheaf F . So (3) is bijective. By 8.3.7 and 8.3.9, (5) is bijective. So (4) is injective. This is true for any constructible sheaf F . So (6) is injective. It then follows that (4) is bijective. This is true for any constructible sheaf F . So (6) is bijective. Using this argument repeatedly, we can prove that the homomorphism Ext2−r (F , µn ) → H r (X, F )∨ is bijective for each r.

Lemma 8.3.10. Let S be a regular scheme pure of dimension 1, U a dense open subset of S, j : U ,→ S the open immersion, A a noetherian ring such that nA = 0 and A is an injective A-module, and F a locally constant constructible sheaf of A-modules on U . Then we have H om(j∗ F , A) ∼ = j∗ H om(F , A) and E xtq (j∗ F , A) = 0 for any q ≥ 1.

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Proof.

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By 8.3.3, we have j ∗ E xtq (j∗ F , A) ∼ = E xtq (F , A) = 0

for all q ≥ 1. The isomorphism

j ∗ H om(j∗ F , A)) ∼ = H om(F , A)

induces a morphism H om(j∗ F , A) → j∗ H om(F , A) whose restriction to U is an isomorphism. To prove the lemma, we need to show that for any s ∈ S − U , we have (E xtq (j∗ F , A))s¯ = 0 for all q ≥ 1, and (H om(j∗ F , A))s¯ → (j∗ H om(F , A))s¯ is an isomorphism. Since S is pure of dimension 1 and U is dense, s is a closed point of X. By 5.9.11, we may replace S by the strict localization of S at s. So we may assume that S is a strictly local trait, s is the closed point of S, and U = {η} is the generic point of S. Let i : s → S and j : η → S be the immersions, and let I = Gal(¯ η /η). We have an exact sequence 0 → j! F → j∗ F → i∗ Fη¯I → 0. It gives rise to a distinguished triangle RH om(i∗ Fη¯I , A) s¯ → RH om(j∗ F , A) s¯ → RH om(j! F , A) s¯ → .

One can verify

RH om(i∗ Fη¯I , A) ∼ = i∗ RH om(Fη¯I , Ri! A). By 8.3.6, we have Ri! A ∼ = A(−1)[−2]. As A is an injective A-module, we have RH om(Fη¯I , Ri! A) ∼ = H om(Fη¯I , A(−1)[−2]) = (Fη¯I )∨ (−1)[−2], where for any A-module M , we set M ∨ = HomA (M, A). We thus have RH om(i∗ Fη¯I , A) ∼ = i∗ (Fη¯I )∨ (−1)[−2].

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One can verify By 8.3.3, we have

RH om(j! F , A) ∼ = Rj∗ RH om(F , A). RH om(F , A) = H om(F , A).

So we have RH om(j! F , A) ∼ = Rj∗ H om(F , A). We thus have a distinguished triangle (Fη¯I )∨ (−1)[−2] → RH om(j∗ F , A) s¯ → Rj∗ H om(F , A) s¯ → . Taking the long exact sequence of cohomology associated to this triangle, we get an exact sequence 0 → (E xt1 (j∗ F , A))s¯ → (R1 j∗ H om(F , A))s¯ → (Fη¯I )∨ (−1) → → (E xt2 (j∗ F , A))s¯ → (R2 j∗ H om(F , A))s¯ → 0 and isomorphisms (E xtq (j∗ F , A))s¯ ∼ = (Rq j∗ Hom(F , A))s¯ for all q 6= 1, 2. By 8.3.5, we have  ∨ I if q = 0,  (Fη¯ ) q ∼ (R j∗ H om(F , A))s¯ = (Fη¯∨ )I (−1) if q = 1,  0 if q 6= 0, 1. It follows that (E xtq (j∗ F , A))s¯ = 0 for any q ≥ 3 and (H om(j∗ F , A))s¯ ∼ = (j∗ H om(F , A))s¯. In the above discussion, we can replace F by its constant subsheaf Fη¯I . Note that j∗ Fη¯I is a constant sheaf on S. So we have E xtq (j∗ Fη¯I , A) = 0 for all q ≥ 1 by 8.3.3. Moreover, we have a commutative diagram (Fη¯∨ )I (−1) ok 0→ (E xt1 (j∗ F ,A))s¯ → (R1 j∗ H om(F ,A))s¯ → (Fη¯I )∨ (−1) → (E xt2 (j∗ F ,A))s¯ →0 ↓

↓

↓

↓

0→ (E xt1 (j∗ Fη¯I ,A))s¯ → (R1 j∗ H om(Fη¯I ,A))s¯ → (Fη¯I )∨ (−1) → (E xt2 (j∗ Fη¯I ,A))s¯ →0 k

ok

k

0

(Fη¯∨ )I (−1)

0

It follows that the homomorphism (R1 j∗ H om(F , A))s¯ → (Fη¯I )∨ (−1) is an isomorphism. So (E xtq (j∗ F , A))s¯ = 0 for q = 0, 1.

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Theorem 8.3.11. Let X be a smooth curve over an algebraically closed field k, U a dense open subset of X, j : U → X the open immersion, F a locally constant constructible sheaf of Z/n-modules on U , and F ∨ = H om(F , Z/n). Then we have a perfect pairing Hcr (X, j∗ F ) × H 2−r (X, j∗ F ∨ (1)) → Z/n for each r. By 8.3.10, the spectral sequence

Proof.

E2pq = H p (X, E xtq (j∗ F , Z/n)) ⇒ Extp+q (j∗ F , Z/n) degenerates and H om(j∗ F , Z/n) ∼ = j∗ F ∨ . So we have Extp (j∗ F , Z/n) ∼ = H p (X, H om(j∗ F , Z/n)) ∼ = H p (X, j∗ F ∨ ). Our assertion then follows from 8.3.1. 8.4

The Functor Rf !

([SGA 4] XVIII 3.1.) Fix a noetherian scheme S. In this section, all schemes are of finite type over S. Let f : X → Y be an S-compactifiable morphism between S-schemes of finite type, and let A be a noetherian torsion ring. In this section, we construct a functor Rf ! : D+ (Y, A) → D+ (X, A) that is right adjoint to the functor Rf! : D(X, A) → D(Y, A). For any sheaf of A-modules F on X, we can write F = lim Fi , −→ i∈I

where (Fi )i∈I is a direct system of constructible sheaves of A-modules on X. Define Cl· (F ) = lim C · (Fi ), −→ i∈I

·

where C (−) denotes the functorial Godement resolution constructed at the end of 5.6. Note that Cl· (F ) is independent of the choice of the direct system (Fi )i∈I . Indeed, suppose F = limj∈J Gj , where Gj are con−→ structible. Let Fi0 (resp. Gj0 ) be the images of the morphisms Fi → F

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(resp. Gj → F ), and let Ki (resp. Lj ) be the kernels of the morphisms Fi → F (resp. Gj → F ). These sheaves are constructible. Using 5.9.8, one can show that for each i (resp. j), there exists i0 ≥ i (resp. j 0 ≥ j) such that the morphism Ki → Ki0 (resp. Lj → Lj 0 ) is zero. It follows that lim C · (Ki ) = 0, lim C · (Lj ) = 0. −→ −→ i

j

So we have lim C · (Fi ) ∼ C · (Fi0 ), = lim −→ −→ i

i

lim C · (Gj ) ∼ C · (Gj0 ). = lim −→ −→ j

j

To prove that Cl· (F ) is independent of the choice of the direct system (Fi )i∈I , it suffices to show lim C · (Fi0 ) ∼ C · (Gj0 ). = lim −→ −→ i

j

For each i, we have Fi0 = ∪j (Fi0 ∩ Gj0 ). It follows that Fi0 = Fi0 ∩ Gj0 for some j, that is, Fi0 ⊂ Gj0 . Similarly, for each j, we can find i such that Gj0 ⊂ Fi0 . Our assertion follows. Lemma 8.4.1. (i) Cl· (F ) is a resolution of F . (ii) For each q, Clq (−) is an exact functor and commutes with the direct limit. (iii) If f : X 0 → X is etale, we have a canonical isomorphism ∼ = ∗ · f Cl (F ) → Cl· (f ∗ F ). Proof. It follows from the definition that Cl· (F ) is a resolution of F , and Clq (−) commutes with the direct limit. Let 0→F →G →H →0 be a short exact sequence of sheaves of A-modules. Write G = limi Gi , −→ where Gi are constructible subsheaves of G . We have F = limi F ∩ Gi , −→ H = limi Gi /F ∩ Gi , and F ∩ Gi , Gi /F ∩ Gi are constructible. We have a −→ short exact sequence 0 → C q (F ∩ Gi ) → C q (Gi ) → C q (Gi /F ∩ Gi ) → 0. Taking direct limit, we get an exact sequence 0 → Clq (F ) → Clq (G ) → Clq (H ) → 0. So Clq (−) is an exact functor. For any S-morphism f : X 0 → X between S-schemes of finite type and any sheaf F on X, we have a canonical morphism f ∗ C · (F ) → C · (f ∗ F ). If f is etale, this is an isomorphism. Using this fact, one proves (iii).

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Let f : X → Y be an S-compactifiable morphism. Fix a compactification j

f¯

X ,→ X → Y, where j is an open immersion and f¯ is a proper S-compactifiable morphism. Let d be an integer such that all the fibers of f¯ have dimensions ≤ d. For any sheaf of A-modules F on X, the complex τ≤2d Cl· (j! F ) is a resolution of j! F by Rf¯∗ -acyclic objects. Indeed, let Fq be the q-th component of τ≤2d Cl· (j! F ). When q 6= 2d, Fq is a direct limit of flasque sheaves. Moreover, we have Rp f¯∗ F2d ∼ = Rp+2d f! F = 0 = Rp+2d f¯∗ j! F ∼ for any p ≥ 1 by 7.4.5. For any complex K of sheaves of A-moduleson X, q denote the complex associated to the bicomplex τ≤2d Cl· (j! K p ) by τ≤2d Cl· (j! K). We then have

p,q

Rf! K ∼ = f¯∗ τ≤2d Cl· (j! K) in D(Y, A). Define a functor f!· from the category of complex of sheaves of A-modules on X to the category of complexes of sheaves of A-modules on Y by f!· (K) = f¯∗ τ≤2d Cl· (j! K). For any sheaf of A-modules F on X, let f!q (F ) be the q-th component of the complex f!· (F ). Then f!q is a functor from the category of sheaves of A-modules on X to the category of sheaves of A-modules on Y . Lemma 8.4.2. (i) For each q, the functor F → f!q (F ) is exact and commutes with the direct limit. We have f!q (F ) = 0 for q 6∈ [0, 2d]. (ii) The functor f!· induces a functor Rf!· : D(X, A) → D(Y, A), and we have a canonical isomorphism Rf!· ∼ = Rf! . Proof. (i) follows from the definition of f!· . As each f!q is exact, the functor f!· maps acyclic complexes to acyclic complexes, and hence defines a functor on the derived category. ! Lemma 8.4.3. For each q, the functor f!q has a right adjoint functor f−q . ! The functor f−q is left exact, transforms injective objects to injective objects, and vanishes if −q 6∈ [−2d, 0].

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! Proof. Given a sheaf of A-modules G on Y , define a presheaf f−q G on X as follows: For any etale X-scheme φ : U → X, let ! (f−q G )(U ) = Hom(f!q AU , G ),

where AU = φ! A. For any etale morphism ψ : V → U , define the restriction ! ! (f−q G )(U ) → (f−q G )(V )

to be the map Hom(f!q AU , G ) → Hom(f!q AV , G ) induced by the canonical morphism AV = φ! ψ! A → φ! A = AU .

! We claim that f−q G is a sheaf. Indeed, if {Uα → U }α is an etale covering of U , then for any sheaf F on X, we have the canonical exact sequence Y Y 0 → F (U ) → F (Uα ) → F (Uα ×U Uβ ). α

α,β

This can be identified with an exact sequence M M 0 → Hom(AU , F ) → Hom AUα , F → Hom AUα ×U Uβ , F . α

α,β

We thus have an exact sequence M M AUα ×U Uβ → AUα → AU → 0. α

α,β

Since

f!q

is exact, we have an exact sequence M q M q f! AUα ×U Uβ → f! AUα → f!q AU → 0, α

α,β

and hence an exact sequence Y Y 0 → Hom(f!q AU , G ) → Hom(f!q AUα , G ) → Hom(f!q AUα ×U Uβ , G ). α

So the sequence

! 0 → (f−q G )(U ) →

α,β

Y Y ! ! (f−q G )(Uα ) → (f−q G )(Uα ×U Uβ ) α

α,β

! f−q G

is exact. This proves is a sheaf. ! Next we define a morphism F → f−q f!q F for any sheaf of A-modules F on X. It suffices to define a homomorphism ! F (U ) → (f−q f!q F )(U ) = Hom(f!q AU , f!q F )

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functorial in U for any etale X-scheme U . Given s ∈ F (U ), it defines a morphism AU → F and hence a morphism f!q AU → f!q F . We assign this ! last morphism to s. The morphism F → f−q f!q F defines a map ! Hom(f!q F , G ) → Hom(F , f−q G ).

By our construction, we have

! Hom(f!q AU , G ) ∼ G) = Hom(AU , f−q

for any etale X-scheme U . If F is constructible, then we can find an exact sequence of the form AV → AU → F → 0

for some etale X-schemes U and V . Since f!q is exact, the sequence f!q AV → f!q AU → f!q F → 0

is exact. We have a commutative diagram

0 → Hom(f!q F , G ) → Hom(f!q AU , G ) → Hom(f!q AV , G ) ↓ ↓ ↓ ! ! ! 0 → Hom(F , f−q G ) → Hom(AU , f−q G ) → Hom(AV , f−q G ).

By the above discussion, the last two vertical arrows are bijective. It follows from the five lemma that ∼ Hom(F , f ! G ) Hom(f q F , G ) = −q

!

if F is constructible. In general, we can write F = limi Fi , where Fi are −→ constructible. Since f!q commutes with the direct limit, we have q Hom(f!q F , G ) ∼ F ), G ) = Hom(f! (lim −→ i i

∼ (f q F ), G ) = Hom(lim −→ ! i i

∼ Hom(f!q Fi , G ) = lim ←− i

! ∼ Hom(Fi , f−q G) = lim ←− i

∼ F , f! G ) = Hom(lim −→ i −q i

! ∼ G ). = Hom(F , f−q

So ! Hom(f!q F , G ) ∼ G) = Hom(F , f−q

! for any sheaf of A-modules F on X. Therefore f−q is right adjoint to f!q . ! Since f!q is exact, f−q transforms injective objects to injective objects. As a right adjoint functor, it is left exact.

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! ! Let dq∗ : f−q−1 → f−q be the functor induced by dq : f!q → f!q+1 by adjunction. We then get a complex of functors f·! with differentials ! ! (−1)q dq∗ : f−q−1 → f−q .

For any complex of sheaves of A-modules L on Y , denote the complex associated to the bicomplex (fq! Lp )p,q by f·! L· . Then for any complex of sheaves of A-modules K on X, we have a canonical isomorphism of complexes of A-modules Hom· (f!· K, L) ∼ = Hom· (K, f·! L). One can show that f·! : K(Y, A) → K(X, A) is an exact functor. Let Rf ! : D+ (Y, A) → D+ (X, A) be its right derived functor. Then Rf ! is right adjoint to Rf! . Indeed, if L is a bounded below complex of injective sheaves of A-modules on Y , then we have Hom(Rf! K, L) ∼ = Hom(f!· K, L) ∼ = H 0 (Hom· (f!· K, L)) ∼ = H 0 (Hom· (K, f·! L)). But f·! L is a complex of injective sheaves of A-modules on X, so we have H 0 (Hom· (K, f·! L)) ∼ = Hom(K, f·! L) ∼ = Hom(K, Rf ! L). Hence Hom(Rf! K, L) ∼ = Hom(K, Rf ! L). We thus have the following: Theorem 8.4.4. Let S be a noetherian scheme, let f : X → Y be an Scompactifiable morphism between S-schemes of finite type, and let A be a noetherian torsion ring. Then the functor Rf ! : D+ (Y, A) → D+ (X, A) is right adjoint to the functor Rf! : D(X, A) → D(Y, A), that is, for any K ∈ ob D(X, A) and L ∈ ob D+ (Y, A), we have a one-to-one correspondence Hom(Rf! K, L) ∼ = Hom(K, Rf ! L) functorial in K and L. Moreover, Rf ! is an exact functor and the following diagram commutes: Hom(Rf! (K[1]), L) ∼ = Hom(K[1], Rf ! L) ∼ = Hom(K, (Rf ! L)[−1]) ∼ ↓∼ = ↓ = ∼ ∼ Hom((Rf! K)[1], L) = Hom(Rf! K, L[−1]) = Hom(K, Rf ! (L[−1])).

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Proposition 8.4.5. Let S be a noetherian scheme, and let f : X → Y be a separated quasi-finite morphism between S-schemes of finite type. (i) The functor f! from the category of sheaves of A-modules on X to the category of sheaves of A-modules on Y has a right adjoint f ! , and Rf ! can be identified with the right derived functor of f ! . (ii) If f is a closed immersion, then f ! can be identified with the functor defined in 5.4. (iii) If f is etale, then f ! can be identified with f ∗ . Proof. (i) follows from the construction of f ! in the case d = 0. (ii) follows from (i) since if f is a closed immersion, then f! = f∗ , and the functor f ! constructed in 5.4 is right adjoint to f∗ . (iii) follows from (i) since if f is etale, then f ∗ is right adjoint to f! . Let f : X → Y be an S-compactifiable morphism between S-schemes of finite type, K ∈ ob D− (X, A) and L ∈ ob D+ (X, A). We define a canonical morphism Rf∗ RH om(K, L) → RH om(Rf! K, Rf! L) as follows. Represent L by a bounded below complex of injective sheaves of A-modules. Then RH om(K, L) ∼ = H om· (K, L). By 5.6.8, H om· (K, L) is a complex of flasque sheaves. So we have Rf∗ RH om(K, L) = f∗ H om· (K, L). Fix a compactification f¯

j

X ,→ X → Y of f and an upper bound d of dimensions of fibers of f¯. For each etale Y -scheme k : V → Y , fix notation by the following commutative diagram f¯V

jV

XV → X V → V k ↓ kX ↓ kX ↓ f¯

j

X→

X → Y,

and let fV = f¯V jV be the base change of f . By 8.4.1 (iii), we have an isomorphism of complexes of functors ∼ =

∗ k ∗ f!· → fV· ! kX .

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It follows that H om· (f!· K, f!· L)(V ) = Hom(k ∗ f!· K, k ∗ f!· L) ∼ Hom(f · k ∗ K, f · k ∗ L). = V! X V! X We have a canonical morphism ∗ ∗ ∗ ∗ H om· (K, L)(XV ) = Hom· (kX K, kX L) → Hom· (fV· ! kX K, fV· ! kX L).

We thus have a morphism of complexes of sheaves f∗ H om· (K, L) → H om· (f!· K, f!· L). It induces a morphism Rf∗ RH om(K, L) → RH om(Rf! K, Rf! L)

in the derived category. Now assume that K ∈ ob D− (X, A) and L ∈ ob D+ (Y, A). Taking the composite of the morphism Rf∗ RH om(K, Rf ! L) → RH om(Rf! K, Rf! Rf ! L) and the morphism RH om(Rf! K, Rf! Rf ! L) → RH om(Rf! K, L)

induced by the canonical morphism Rf! Rf ! L → L, we get a morphism Rf∗ RH om(K, Rf ! L) → RH om(Rf! K, L).

Theorem 8.4.6. Let S be a noetherian scheme, let f : X → Y be an S-compactifiable morphism between S-schemes of finite type, let A be a noetherian torsion ring, and let K ∈ ob D− (X, A) and L ∈ ob D+ (Y, A). The morphism Rf∗ RH om(K, Rf ! L) → RH om(Rf! K, L) defined above is an isomorphism. Proof. Represent L by a bounded below complex of injective sheaves of A-modules on Y . We have Rf∗ RH om(K, Rf ! L) ∼ = f∗ H om· (K, f·! L), ∼ H om· (f · K, L). RH om(Rf! K, L) = !

Let us prove that for any etale Y -scheme k : V → Y , that the morphism H om· (K, f·! L) (XV ) → H om· (f!· K, L) (V )

is a quasi-isomorphism, that is, the morphism

∗ ∗ ! Hom· (kX K, kX f· L) → Hom· (k ∗ f!· K, k ∗ L)

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is a quasi-isomorphism. Without loss of generality, let us prove that ∗ ∗ ! H 0 Hom· (kX K, kX f· L) → H 0 Hom· (k ∗ f!· K, k ∗ L) is an isomorphism, that is,

∗ ∗ ! K, kX f· L) ∼ Hom(kX = Hom(k ∗ f!· K, k ∗ L).

Here Hom(−, −) denotes the spaces of homotopy classes of morphisms of complexes. When V = Y , this follows from the adjointness of f!· and f·! . ∗ ! For general V , we need to introduce a morphism kX f· → fV! · k ∗ . We have an isomorphism of complexes of functors ∼ =

∗ . k ∗ f!· → fV· ! kX

For any q, we get a morphism k! fVq ! → f!q kX! by taking the composite ∼ =

∗ k! fVq ! → k! fVq ! kX kX! → k! k ∗ f!q kX! → f!q kX! .

∗ ! By the adjointness of the pairs of functors (k! , k ∗ ), (kX! , kX ), (fVq ! , fV,−q ) q ! and (f! , f−q ), it induces a morphism ∗ ! ! kX f−q → fV,−q k∗ .

In this way, we get a morphism of complex of functors ∗ ! kX f· → fV! · k ∗ .

It induces an isomorphism in the derived category since k! fV· ! → f!· kX! induces an isomorphism in the derived category. Since L is a bounded below complex of injective sheaves, ∗ ! kX f· L → fV! · k ∗ L

is an isomorphism in K(XV , A) by 6.2.7. Consider the diagram ∼ =

∗ ∗ ! Hom(kX K,kX f· L)

→

∗ Hom(kX K,fV! · k∗ L)

↓

(1)

↓

(2)&∼ =

∗ ∗ ! Hom(fV· ! kX K,fV· ! kX f· L)

∼ =

→

∗ Hom(fV· ! kX K,fV· ! fV! · k∗ L)

→

∼ =↓

(3)

↓∼ =

(4)

↓∼ =

∗ ! Hom(k∗ f!· K,fV· ! kX f· L)

→

∼ =

Hom(k∗ f!· K,fV· ! fV! · k∗ L)

→

Hom(k∗ f!· K,k∗ L),

(5)

%

-∼ =

∗ Hom(fV· ! kX K,k∗ L)

Hom(k∗ f!· K,k∗ f!· f·! L)

where Hom(−, −) denotes the space of homotopy classes of morphisms of complexes, the horizontal arrows in the squares (1) and (3) are induced

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∗ ! by the isomorphism kX f· L → fV! · k ∗ L, the vertical arrows in the squares (3) and (4) and the left slant arrow in the triangle (5) are induced by the ∼ = ∗ isomorphism k ∗ f!· K → fV· ! kX K, the horizontal arrows in the square (4) and the right slant arrow in the triangle (5) are induced by the canonical morphisms fV· ! fV! · → id and f!· f·! → id, respectively. By the adjointness of (fV· ! , fV! · ), the slant arrow in the triangle (2) is an isomorphism. The morphism ∗ ∗ ! Hom(kX K, kX f· L) → Hom(k ∗ f!· K, k ∗ L)

defined at the beginning is the composite of those morphisms on the left and the bottom part of the boundary of the above diagram. (In the composite, we take the inverse of the left slant arrow in the triangle (5).) On the other hand, the morphisms on the top and the right part of the boundary of the ∗ ∗ ! above diagram are isomorphisms. So to prove that Hom(kX K, kX f· L) → ∗ · ∗ Hom(k f! K, k L) is an isomorphism, it suffices to show that the above diagram commutes. It is clear that (1), (2), (3) and (4) commute. To prove that (5) commutes, it suffices to show that the following diagram commutes: k∗ (adj)

k ∗ f!· f·! L → k∗ L ∼ (6) ↑adj = ↓ ∼ = · ∗ ! · fV ! kX f· L → fV ! fV! · k ∗ L. Recall that we define k! fV· ! → f!· kX! as the composite ∼ =

∗ k! fV· ! → k! fV· ! kX kX! → k! k ∗ f!· kX! → f!· kX! . ∗ ! We then define the morphism kX f· → fV! · k ∗ as the composite ∗ ! ∗ ! ∗ ! kX f· → fV! · k ∗ k! fV· ! kX f· → fV! k ∗ f!· kX! kX f· → fV! k ∗ . ∗ ! It follows that kX f· → fV! · k ∗ is the composite of those morphisms on the left part of the boundary of the following diagram. ∗ ! kX f· . (7) ↓ ∗ ∗ ! ∗ ! fV! · k ∗ k! fV· ! kX kX! kX f· → fV! · k ∗ k! fV· ! kX f· ↓ ↓ ∗ ! fV! · k ∗ k! k ∗ f!· kX! kX f· → fV! · k ∗ k! k ∗ f!· f·! & ↓ fV! · k ∗

& ∗ ! ← fV! · fV· ! kX f· ↓ ← fV! · k ∗ f!· f·! (8) . .

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In the above diagram, the triangles (7) and (8) commute because the following two diagrams commute. ∗ kX k

∗ kX

adj

→ (9) ∗ kX (adj)

←

adj

∗ ∗ (kX kX! )kX k

k ∗ → (k ∗ k! )k ∗ k (10) k

∗ ∗ kX (kX! kX ),

k∗

k∗ (adj)

←

k ∗ (k! k ∗ ).

(The commutativity of (9) can be seen as follows: By the adjointness of the ∗ pair (kX! , kX ), the composite of the morphisms adj

∗ ∗ ∗ ∗ ∗ kX → (kX kX! )kX = kX (kX! kX )

∗ kX (adj)

→

∗ kX

corresponds to the composite of the morphisms adj

id

∗ ∗ kX! kX → kX! kX → id.

∗ ∗ The second composite is adj : kX! kX → id, and it corresponds to id : kX → ∗ ∗ kX by the adjointness of (kX! , kX ). Similarly (10) commutes.) It is clear that the other parts of the above diagram commute. So the morphism ∗ ! kX f· → fV! · k ∗ is also the composite ∼ =

∗ ! ∗ ! kX f → fV! · fV· ! kX f· → fV! · k ∗ f!· f·! → fV! · k ∗ .

The commutativity of (6) then follows from the commutativity of the following diagram: k ∗ f!· f·! .

∗ ! id fV· ! kX f· →

∼ =

& (11)

∗ ! fV· ! kX f·

→ ∼ =

→

k∗ ↑

k ∗ f!· f·!

-

fV· ! fV! · k ∗ .

↑ ↑ % ∗ ! fV· ! fV! · fV· ! kX f· → fV· ! fV! · k ∗ f!· f·!

In this last diagram, the triangle (11) commutes because the following diagram commutes fV· ! k fV· !

fV· (adj) !

→

adj

←

fV· ! (fV! · fV· ! ) k

(fV· ! fV! · )fV· ! .

Theorem 8.4.7. Let S be a noetherian scheme, f : X → Y an Scompactifiable morphism between S-schemes of finite type, A a noetherian ring, K ∈ ob D− (Y, A) and L ∈ D+ (Y, A). Then we have a canonical isomorphism ∼ =

RH om(f ∗ K, Rf ! L) → Rf ! RH om(K, L).

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Proof. Roughly speaking, the proof goes as follows. D− (X, A), by the projection formula 7.4.7, we have

For any M ∈

∼ =

∗ Rf! M ⊗L A K → Rf! (M ⊗ f K).

Hence ∼ =

∗ L Hom(Rf! (M ⊗L A f K), L) → Hom(Rf! M ⊗A K, L).

By the proof of 6.4.7, we have ∗ L ∗ ! ∼ Hom(Rf! (M ⊗L A f K), L) = Hom(M ⊗A f K, Rf L) ∼ = Hom(M, RH om(f ∗ K, Rf ! L)), Hom(Rf! M ⊗L K, L) ∼ = Hom(Rf! M, RH om(K, L)) A

∼ = Hom(M, Rf ! RH om(K, L)).

So we have a one-to-one correspondence ∼ =

Hom(M, RH om(f ∗ K, Rf ! L)) → Hom(M, Rf ! RH om(K, L)). ∼ =

It induces an isomorphism RH om(f ∗ K, Rf ! L) → Rf ! RH om(K, L). More explicitly, we construct the isomorphism as follows. Represent K by a bounded above complex of flat sheaves. We have · Rf! M ⊗L A K = f! M ⊗A K,

∗ ∗ ∼ · Rf! (M ⊗L A f K) = f! (M ⊗A f K).

We first construct a functorial morphism of complexes f!· M ⊗A K → f!· (M ⊗A f ∗ K). Let j

f¯

X ,→ X → Y

be a compactification of f , where j is an open immersion and f¯ is a proper S-compactifiable morphism. For sheaves of A-modules H1 and H2 on X, we construct a morphism C · (H1 ) ⊗A H2 → C · (H1 ⊗A H2 ) as follows. Let PX be the set of geometric points in X that we use to construct the Godement resolution. For any geometric point t : Spec k → X in PX , we have a canonical morphism t∗ t∗ H1 ⊗A H2 → t∗ (t∗ H1 ⊗A t∗ H2 ). Taking direct product over all t ∈ PX , we can define a morphism C 0 (H1 ) ⊗A H2 → C 0 (H1 ⊗A H2 ).

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Note that the following diagram commutes: H1 ⊗A H2 → C 0 (H1 ) ⊗A H2 k ↓ H1 ⊗A H2 → C 0 (H1 ⊗A H2 ). Suppose we have defined C q (H1 ) ⊗A H2 → C q (H1 ⊗A H2 ) for all q ≤ n such that the following diagram commute: C q−1 (H1 ) ⊗A H2 → C q (H1 ) ⊗A H2 ↓ ↓ C q−1 (H1 ⊗A H2 ) → C q (H1 ⊗A H2 ). These morphisms induce a morphism coker (C n−1 (H1 ) → C n (H1 )) ⊗A H2

→ coker (C n−1 (H1 ⊗A H2 ) → C n (H1 ⊗A H2 )).

We define C n+1 (H1 ) ⊗A H2 → C n+1 (H1 ⊗A H2 ) as the composite of the following morphisms: C n+1 (H1 ) ⊗A H2 = C 0 (coker (C n−1 (H1 ) → C n (H1 ))) ⊗A H2

→ C 0 (coker (C n−1 (H1 ) → C n (H1 )) ⊗A H2 )

→ C 0 (coker (C n−1 (H1 ⊗A H2 ) → C n (H1 ⊗A H2 ))) = C n+1 (H1 ⊗A H2 ).

For any sheaf of A-modules F on X and any sheaf of A-modules G on Y , write F = limi Fi and G = limj Gj , where Fi and Gj are constructible −→ −→ sheaves of A-modules. We define a morphism of complexes f!· F ⊗A G → f!· (F ⊗A f ∗ G )

as the composite of the following morphisms: f!· F ⊗A G ∼ C · (j! Fi )) ⊗A lim Gj = f¯∗ τ≤2d (lim −→ −→ i j → f¯∗ τ≤2d (lim C · (j! Fi )) ⊗A f¯∗ (lim Gj ) −→ −→ i j → f¯∗ τ≤2d lim C · (j! Fi ) ⊗A f¯∗ (lim Gj ) −→ −→ i

j

∼ (C · (j! Fi ) ⊗A f¯∗ Gj ) = f¯∗ τ≤2d lim −→ i,j

→ f¯∗ τ≤2d lim C · (j! Fi ⊗A f¯∗ Gj ) −→ i,j

→ f¯∗ τ≤2d lim C · (j! (Fi ⊗A f ∗ Gj )) −→ i,j

∼ = f!· (F ⊗A f ∗ G ),

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where d is an upper bound for the dimensions of fibers of f¯. We then use this morphism to define f!· M ⊗A K → f!· (M ⊗A f ∗ K). For any q and any sheaf of A-modules H on S, the morphism f!q F ⊗A G → f!q (F ⊗A f ∗ G )

induces a homomorphism We have

Hom(f!q (F ⊗ f ∗ G ), H ) → Hom(f!q F ⊗ G , H ).

! Hom(f!q (F ⊗A f ∗ G ), H ) ∼ H) = Hom(F ⊗A f ∗ G , f−q ∗ ∼ = Hom(F , H om(f G , f ! H )), −q

q Hom(f!q F ⊗A G , H ) ∼ = Hom(f! F , H om(G , H )) ∼ = Hom(F , f ! H om(G , H )). −q

So we have a homomorphism ! ! Hom(F , H om(f ∗ G , f−q H )) → Hom(F , f−q H om(G , H )).

It gives rise to a morphism

! ! H om(G , H ). H ) → f−q H om(f ∗ G , f−q

From this, we can define a morphism of complexes

H om· (f ∗ K, f·! L) → f·! H om· (K, L).

Represent K by a bounded above complex of flat sheaves, and L by a bounded below complex of injective sheaves. We have Rf! M ⊗L K ∼ = f · M ⊗A K, A

!

∗ ∗ ∼ · Rf! (M ⊗L A f K) = f! (M ⊗A f K), RH om(f ∗ K, Rf ! L) ∼ = H om· (f ∗ K, f ! L), ·

Rf ! RH om(K, L) ∼ = f·! H om· (K, L).

We thus get a morphism RH om(f ∗ K, Rf ! L) → Rf ! RH om(K, L)

in the derived category. The following diagram commutes: ∼ Hom(Rf! M ⊗L = Hom(M, Rf ! RH om(K, L)) A K, L) ok ok H 0 (Hom· (f!· M ⊗A K, L)) ∼ = H 0 (Hom· (M, f·! H om· (K, L))) ↑ ↑ · · · 0 ∗ 0 ∼ H (Hom (f! (M ⊗A f K), L)) = H (Hom (M, H om· (f ∗ K, f·! L))) ok ok ∗ ∗ ∼ Hom(Rf! (M ⊗L f K), L) Hom(M, RH om(f K, Rf ! L)). = A

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∗ ∼ Since Rf! M ⊗L A K = Rf! (M ⊗A f K), the left vertical arrow in the above diagram is bijective. So we have an isomorphism ∼ =

Hom(M, RH om(f ∗ K, Rf ! L)) → Hom(M, Rf ! RH om(K, L)).

We need to prove that RH om(f ∗ K, Rf ! L) → Rf ! RH om(K, L) is an isomorphism. Let RH om(f ∗ K, Rf ! L) → Rf ! RH om(K, L) → ∆ →

be a distinguished triangle. It suffices to show that ∆ is acyclic. By 6.1.1 (ii), we have Hom(M, ∆) = 0 −

for any M ∈ ob D (X, A). Representing ∆ by a bounded below complex of injective sheaves and take M = G [q] for any q and any sheaf of A-modules G on X, we see that the complex Hom· (G , ∆) is acyclic. By 8.4.8 below, ∆ is acyclic. d0

d

Lemma 8.4.8. Let F 0 → F → F 00 be morphisms of sheaves. If for any sheaf G , the sequence Hom(G , F 0 ) → Hom(G , F ) → Hom(G , F 00 ) d0

d

is exact, then F 0 → F → F 00 is exact. Proof.

The sequence Hom(F 0 , F 0 ) → Hom(F 0 , F ) → Hom(F 0 , F 00 )

is exact. The morphism idF 0 is mapped to 0 in Hom(F 0 , F 00 ) under the composite of the above homomorphisms. So we have dd0 = 0. The sequence Hom(ker d, F 0 ) → Hom(ker d, F ) → Hom(ker d, F 00 )

is exact. The inclusion i : ker d ,→ F is mapped to 0 in Hom(ker d, F 00 ). So there exists a morphism φ : ker d → F 0 such that d0 φ = i. This implies that ker d ∼ = im d0 . Proposition 8.4.9. Let S be a noetherian scheme. Consider a Cartesian diagram g0

X ×Y Y 0 → X f0 ↓ ↓f g Y0 → Y. Suppose that X, Y , and Y 0 are S-schemes of finite type, and f is Scompactifiable. For any K ∈ ob D+ (Y 0 , A), there exists a canonical isomorphism ∼ =

Rg∗0 Rf 0! K → Rf ! Rg∗ K.

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Proof.

For any L ∈ ob D(X, A), we have a canonical morphism ∼ =

g ∗ Rf! L → Rf!0 g 0∗ L. It induces a one-to-one correspondence ∼ =

Hom(Rf!0 g 0∗ L, K) → Hom(g ∗ Rf! L, K). We have Hom(Rf!0 g 0∗ L, K) ∼ = Hom(g 0∗ L, Rf 0! K) ∼ = Hom(L, Rg∗0 Rf 0! K), Hom(g ∗ Rf! L, K) ∼ = Hom(Rf! L, Rg∗ K) ∼ = Hom(L, Rf ! Rg∗ K). So we have a one-to-one correspondence ∼ =

Hom(L, Rg∗0 Rf 0! K) → Hom(L, Rf !Rg∗ K). It gives rise to an isomorphism ∼ =

Rg∗0 Rf 0! K → Rf ! Rg∗ K.

Corollary 8.4.10. Let S be a noetherian scheme. Consider a Cartesian diagram g0

X ×Y Y 0 → X f0 ↓ ↓f g Y0 → Y. Suppose that X, Y , and Y 0 are S-schemes of finite type, and f is Scompactifiable. For any K ∈ ob D− (Y 0 , A) and L ∈ ob D+ (Y 0 , A), we have a canonical isomorphism ∼ =

Rg∗0 RH om(f 0∗ K, Rf 0! L) → Rf ! Rg∗ RH om(K, L). Proof.

Follows from 8.4.7 and 8.4.9.

Lemma 8.4.11. Let X be a scheme and let A be a ring. Suppose F = L j j! A, where j goes over a set of etale morphisms j : U → X. Then for any flasque sheaf of A-modules G on X, we have ExtqA (F , G ) = 0 and E xtqA (F , G ) = 0 for all q ≥ 1. Proof. Since E xtqA (F , G ) is the sheaf associated to the presheaf V 7→ ExtqA (F |V , G |V ), it suffices to treat Ext. Let I · be a resolution of G by

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injective sheaves of A-modules. For any q ≥ 1, we have M q q · ∼ ExtA (F , G ) = H HomA j! A, I ∼ = Hq ∼ = Hq ∼ =

Y

Y j

Y j

j

HomA (j! A, I ) ·

Γ(U, I · )

q

H (U, G )

j

= 0.

Let X be a scheme, let A be a ring, and let B be an A-algebra. Consider the functor ρ : K(X, B) → K(X, A) that maps each complex of sheaves of B-modules on X to the same complex considered as a complex of sheaves of A-modules on X. Then ρ transforms quasi-isomorphisms to quasi-isomorphisms. So it induces a functor ρ : D(X, B) → D(X, A) on the derived categories. Lemma 8.4.12. Let X be a scheme, A a ring, B an A-algebra, and ρ : D(X, B) → D(X, A) the functor defined above. For any K ∈ ob D− (X, A) and L ∈ ob D+ (X, B), we have ∼ HomD(X,B) (K ⊗L A B, L) = HomD(X,A) (K, ρ(L)). Proof. We may assume that L is a bounded below complex of injective sheaves of B-modules, and K is a bounded above complex whose compoL nents are of the form j j! A, where j goes over a set of etale morphisms to X. We then have · ∼ 0 HomD(X,B) (K ⊗L A B, L) = H (Hom (K ⊗A B, L)) ∼ = H 0 (Hom· (K, ρ(L))).

To prove our assertion, it suffices to show H 0 (Hom· (K, ρ(L))) ∼ = HomD(X,A) (K, ρ(L)).

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Let I ·· be a Cartan–Eilenberg resolution of ρ(L). For each q, I ·q is a resolution of ρ(Lq ) by injective sheaves of A-modules. By 8.4.11, we have Extv (K −u , ρ(Lq )) = 0 for all v ≥ 1 and all u, q. It follows that Hom(K −u , ρ(Lq )) v −u ·q H Hom(K , I ) = 0

if v = 0, if v = 6 0.

So the biregular spectral sequence

E1uv = H v (Hom(K −u , I ·q )) ⇒ H u+v (Hom(K · , I ·q )) degenerates at the level E1·· , and we have H u (Hom(K · , ρ(Lq ))) ∼ = H u (Hom(K · , I ·q )). Consider the bicomplexes C1·· and C2·· defined by C1pq = Hom(K −p , ρ(Lq )), M C2pq = Hom(K −u , I vq ). u+v=p

We have a morphism of bicomplexes C1·· → C2·· . It induces a morphism between the following two spectral sequences E1pq = H q (C1·p ) ⇒ H p+q (C1·· ),

E1pq = H q (C2·p ) ⇒ H p+q (C2·· ).

We have just shown that ·p H q (C1·p ) ∼ = H q (C2 ).

It follows that H p (C1·· ) ∼ = H p (C2·· ) for all p. We have H 0 (C1·· ) = H 0 (Hom· (K, ρ(L))), H 0 (C2·· ) = HomD(X,A) (K, ρ(L)). We thus have H 0 (Hom· (K, ρ(L))) ∼ = HomD(X,A) (K, ρ(L)).

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Proposition 8.4.13. Let S be a noetherian ring, f : X → Y an Scompactifiable morphism between S-schemes of finite type, A a noetherian ring, B a noetherian A-algebra, L ∈ ob D+ (Y, B), ρ : D(X, B) → D(X, A) and ρ : D(Y, B) → D(Y, A) the functors defined above. Then we have a canonical isomorphism ∼ =

ρRf ! L → Rf ! ρL. Proof.

By 7.4.7, we have L ∼ Rf! K ⊗L A B = Rf! (K ⊗A B)

for any K ∈ ob D− (X, A). Combined with 8.4.12, we have

! HomD(X,A) (K, ρRf ! L) ∼ = HomD(X,B) (K ⊗L A B, Rf L) ∼ HomD(Y,B) (Rf! (K ⊗L B), L) = A

∼ = HomD(Y,B) (Rf! K ⊗L A B, L) ∼ HomD(Y,A) (Rf! K, ρL) = ∼ = HomD(X,A) (K, Rf ! ρL).

It follows that we have an isomorphism ρRf ! L ∼ = Rf ! ρL. More explicitly, we construct the isomorphism as follows. As in the proof of 8.4.7, we can construct a morphism of complexes f!· F ⊗A B → f!· (F ⊗A B) for any sheaf of A-modules F on X. We then use this morphism to define f!· K ⊗ B → f!· (K ⊗ B) for any complex K of sheaves of A-modules on X. Moreover, for any q and any sheaf of B-modules H on Y , the morphism f!q F ⊗A B → f!q (F ⊗A B) induces a homomorphism Hom(f!q (F ⊗A B), H ) → Hom(f!q F ⊗A B, H ). We have ! Hom(f!q (F ⊗A B), H ) ∼ H) = Hom(F ⊗A B, f−q ! ∼ = Hom(F , ρf H ), −q

q Hom(f!q F ⊗A B, H ) ∼ = Hom(f! F , ρH ) ∼ = Hom(F , f ! ρH ). −q

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So we have a homomorphism ! ! Hom(F , ρf−q H ) → Hom(F , f−q ρH ).

It gives rise to a morphism ! ! ρf−q H → f−q ρH .

For any L ∈ ob D+ (Y, B), let L → I and ρI → J be quasi-isomorphisms such that I (resp. J) is a bounded below complex of injective sheaves of B-modules (resp. A-modules) on Y . We define a morphism ρRf ! L → Rf ! ρL in the derived category to be the composite ρRf ! L ∼ = ρf·! I → f·! ρI → f·! J ∼ = Rf ! ρL. Represent K by a bounded above complex whose components are of the L form j j! A, where j goes over a set of etale morphisms to X. The following diagram commutes: Hom(Rf! K ⊗L A B, L) ↑ H 0 (Hom· (f!· K ⊗A B, I)) ↑ · · 0 H (Hom (f! (K ⊗A B), I)) ok Hom(Rf! (K ⊗L A B), L)

∼ =

Hom(K, Rf ! ρL) ↑ ∼ = H 0 (Hom· (K, f·! ρI)) ↑ · 0 ∼ = H (Hom (K, ρf·! I)) ok ∼ = Hom(K, ρRf ! L).

∼ Since Rf! K ⊗L A B = Rf! (K ⊗A B), the composite of vertical arrows on the left in the above diagram is bijective. So the morphism ρRf ! L → Rf ! ρL induces a bijection ∼ =

Hom(K, ρRf ! L) → Hom(K, Rf ! ρL) for any K ∈ ob D− (X, A). As in the proof of 8.4.7, this implies that ρRf ! L ∼ = Rf ! ρL. 8.5

Poincar´ e Duality

([SGA 4] XVIII 3.2.) Let S be a noetherian scheme, n an integer invertible on S, A a noetherian ring with nA = 0, and f : X → Y a smooth S-compactifiable morphism

etale

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pure of relative dimension d between S-schemes of finite type. Fix a compactification f¯

X ,→ X → Y of f . Consider a commutative diagram k

U → X g ↓ ↓f j

V → Y

such that j and k are etale. We have morphisms of complexes f!· (k! Z/n)

→ ∼ = ∼ =

j! (Trg )

→

H 2d (f!· (k! Z/n))[−2d] R2d f! k! Z/n[−2d] j! R2d g! Z/n[−2d] j! Z/n(−d)[−2d].

For any bounded below complex of sheaves of A-modules K on Y , the composite of the above morphisms induces a homomorphism Hom(j! Z/n(−d)[−2d], K) → Hom(f!· k! Z/n, K). We thus get a morphism of complexes K(V )(d)[2d] → (f·! K)(U ). It induces a morphism tf : f ∗ K(d)[2d] → Rf ! K in the derived category D(X, A). Lemma 8.5.1. Notations as above. The following diagram commutes: Rf! f ∗ K(d)[2d] Trf &

Rf! (tf )

→ K

Rf! Rf ! K. . adj

Proof. Represent K by a bounded below complex of injective sheaves of A-modules on S. Consider a Cartesian diagram j0

U = V ×Y X → X f0 ↓ ↓f j

V → Y,

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where j is etale. By 8.2.4 (i), the following diagram commutes: ∼ =

∼ =

∼ =

R2d f! f ∗ j! Z/n(d) → R2d f! Z/n(d) ⊗Z/n j! Z/n → j! j ∗ R2d f! Z/n(d) → j! R2d f!0 Z/n(d) Trf ↓ j! Z/n

j! j ∗ (Trf ) ↓

Trf ⊗ id ↓ ∼ =

→

Z/n ⊗Z/n j! Z/n

∼ =

→

j! j ∗ Z/n(d)

j! (Trf 0 ) ↓ ∼ =

→

j! Z/n.

It follows that the top square in the following diagram commutes: Hom· (j! Z/n, K) k

Hom· (j! Z/n, K) ↓

Trf

→ Hom· (f!· f ∗ j! Z/n(d)[2d], K) k

Trf 0

→ Hom· (f!· f ∗ j! Z/n(d)[2d], K) ok tf

Hom· (f ∗ j! Z/n, f ∗ K) → Hom· (f ∗ j! Z/n(d)[2d], f·! K), where the first horizontal arrow is induced by the composite Trf

f!· f ∗ j! Z/n(d)[2d] → R2d f! f ∗ j! Z/n(d) → j! Z/n, and the second horizontal arrow is induced by the composite j! (Trf 0 ) f!· f ∗ j! Z/n(d)[2d] → R2d f! f ∗ j! Z/n(d) ∼ = j! R2d f!0 Z/n(d) → j! Z/n.

One can verify that the bottom square in the above diagram commutes. So we have a commutative diagram Hom· (j! Z/n, K) ↓

Trf

→ Hom· (f!· f ∗ j! Z/n(d)[2d], K) ↓ tf

Hom· (f ∗ j! Z/n, f ∗ K) → Hom· (f ∗ j! Z/n(d)[2d], f·! K). This is true for any etale Y -scheme j : V → Y . One deduces from this first for any sheaf of A-modules L on Y , and then for any complex of sheaves of A-modules L on Y , that the following diagram commutes: Hom· (L, K) ↓

Trf

→ Hom· (f!· f ∗ L(d)[2d], K) ↓ tf

Hom· (f ∗ L, f ∗ K) → Hom· (f ∗ L(d)[2d], f·! K). Taking L = K and calculating the images of id ∈ Hom(L, K) under morphisms in the above commutative diagram, we see that Trf corresponds to tf by the adjointness of (Rf! , Rf ! ). The main result of this section is the following theorem:

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Theorem 8.5.2. Let S be a noetherian scheme, n an integer invertible on S, A a noetherian ring such that nA = 0, and f : X → Y a smooth Scompactifiable morphism pure of relative dimension d between S-schemes of finite type. Then the morphism tf : f ∗ L(d)[2d] → Rf ! L is an isomorphism for any L ∈ ob D+ (Y, A). For any K ∈ ob D(X, A), the morphism Trf : Rf! f ∗ L(d)[2d] → L induces a one-to-one correspondence ∼ =

Hom(K, f ∗ L(d)[2d]) → Hom(Rf! K, L),

φ 7→ Trf ◦ Rf! (φ).

Before proving the theorem, we deduce some of its consequences. Corollary 8.5.3 (Poincar´ e Duality). Let X be a smooth compactifiable scheme pure of dimension d over a separably closed field, let n be an integer invertible on X, and let A be a noetherian ring such that nA = 0 and A is an injective A-module. For any sheaf of A-modules F on X, we have an isomorphism Ext2d−q (F , A(d)) ∼ = HomA (Hcq (X, F ), A) A for any q. If F is locally constant, then we have an isomorphism H 2d−q (X, H omA (F , A(d))) ∼ = HomA (Hcq (X, F ), A). Proof.

We have

Hom(F , f ∗ A(d)[2d − i]) ∼ (F , A(d)), = Ext2d−i A −i ∼ HomA (H i (X, F ), A). Hom(Rf! F , A[−i]) = Ext (RΓc (X, F ), A) = A

c

Here for the second equation, we use the assumption that A is an injective A-module. By 8.5.2, we have Hom(F , f ∗ A(d)[2d − i]) ∼ = Hom(Rf! F , A[−i]). Our assertion follows. If F is locally constant, then by 8.3.3, we have E xtqA (F , A) = 0 for any q ≥ 1. It follows that

ExtqA (F , A) ∼ = H q (X, H omA (F , A)).

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Lemma 8.5.4. Let A be a ring. For any A-module M , let M ∨ = HomA (M, A). If A is an injective A-module and M has finite presentation, then the canonical homomorphism M → M ∨∨ is an isomorphism. Proof. Since A is an injective A-module, the functor M → M ∨ is an exact functor. Suppose M has finite presentation. Then we can find an exact sequence Am → An → M → 0. The following diagram commutes: Am → An → M → 0 ↓ ↓ ↓ (Am )∨∨ → (An )∨∨ → M ∨∨ → 0. The first two vertical arrows are isomorphisms. By the five lemma, M → M ∨∨ is an isomorphism. Proposition 8.5.5 (Weak Lefschetz Theorem). Let k be a separably closed field, n an integer relatively prime to the characteristic of k, A a noetherian ring such that nA = 0 and A is an injective A-module, X a N closed subscheme of PN k , and H a hyperplane of Pk . Suppose that X −X ∩H is smooth. For any sheaf of A-modules F on X such that F |X−X∩H is locally constant and constructible, the canonical homomorphism H q (X, F ) → H q (X ∩ H, F |X∩H ) is bijective for i < dim X − 1 and injective for i = dim X − 1. Proof.

By 7.5.2, we have H q (X − X ∩ H, H omA (F , A(d))) = 0

for any q > dim X. By 8.5.3 and 8.5.4, we have Hcq (X − X ∩ H, F ) = 0 for any q < dim X. Our assertion follows from the long exact sequence · · · → Hci (X − X ∩ H, F ) → H i (X, F ) → H i (X ∩ H, F |X∩H ) → · · · .

Corollary 8.5.6 (Relative Purity Theorem). Consider a commutative diagram U

j

i

,→ X ← Z, f |U & ↓ f . f |Z Y

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where i is a closed immersion and U = X − Z. Let c = codim(Z, X), let A be a noetherian ring such that nA = 0 for some integer n invertible on S, and let F be a sheaf of A-modules on X. Suppose that S is noetherian, f and f |Z are smooth S-compactifiable morphisms between S-schemes of finite type, c ≥ 1, and locally with respect to the etale topology, and F is isomorphic to f ∗ G for some sheaf of A-modules G on Y . Then we have ∗ i F (−c) if q = 2c, q ! ∼ R iF = 0 if q 6= 2c,  if q = 0, F Rq j∗ j ∗ F ∼ = i∗ i∗ F (−c) if q = 2c − 1,  0 if q 6= 0, 2c − 1, q q ∗ ∼ R f∗ F = R (f |U )∗ (j F ) if 0 ≤ q ≤ 2c − 2 and we have a long exact sequence 0 → R2c−1 f∗ F → R2c−1 (f |U )∗ j ∗ F →

(f |Z )∗ i∗ F (−c)

→ ···

→ Rq−1 f∗ F → Rq−1 (f |U )∗ j ∗ F → Rq−2c (f |Z )∗ i∗ F (−c) → · · · .

Proof. The problem is local with respect to the etale topology on X. We may assume F = f ∗ G . We have Rf! Ri! ∼ = R(f |Z )! . It follows that R(f |Z )! ∼ = Ri! Rf ! . Let d1 and d2 be the relative dimensions of f and f |Z , respectively. We have c = d1 − d2 , and by 8.5.2, we have Rf ! G ∼ = f ∗ G (d1 )[2d1 ], R(f |Z )! G ∼ = (f |Z )∗ G (d2 )[2d2 ]. So we have Ri! f ∗ G (d1 )[2d1 ] ∼ = (f |Z )∗ G (d2 )[2d2 ]. Hence we have an isomorphism Ri! F ∼ = i∗ F (−c)[−2c]. We have an exact sequence 0 → i∗ i! F → F → j∗ j ∗ F → i∗ R1 i! F → 0

and Rq j∗ j ∗ F = i∗ Rq+1 i! F for any q ≥ 1. This implies the equations in the corollary. The long exact sequence comes from the spectral sequence E2pq = Rp f∗ Rq j∗ (j ∗ F ) ⇒ Rp+q (f |U )∗ j ∗ F .

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To prove 8.5.2, we need the following lemma. Lemma 8.5.7. Let S be a noetherian scheme, let f : X → Y be a smooth S-compactifiable morphism pure of relative dimension d, and let n be an integer invertible on S. Then for any x ∈ X, there exists an etale neighborhood V of f (¯ x) in Y and an etale neighborhood U of x ¯ in X ×Y V such that the canonical morphism Rq fV0 ! Z/n → Rq fV ! Z/n vanishes for any q < 2d and the restriction of TrfV : R2d fV! Z/n(d) → Z/n to the image of the canonical morphism R2d fV0 ! Z/n → R2d fV ! Z/n is an isomorphism, where the notation is given by the following commutative diagram: j

U fV0

→ X ×Y V → X & fV ↓ ↓f V → Y.

We give the proof of 8.5.7 after the proof of 8.5.2. Proof of 8.5.2. It suffices to show that tf is an isomorphism. By 8.4.13, we may assume A = Z/n. For any x ∈ X, let S be the category defined as follows: Objects in S are triples (U, V, g), where k : U → X is an etale neighborhood of x¯, j : V → Y is an etale neighborhood of f (¯ x), and g : U → V is a morphism satisfying f k = jg; morphisms in S from an object (U, V, g) to an object (U 0 , V 0 , g 0 ) is a pair (φ, ψ) such that φ : U → U 0 is an X-morphism, ψ : V → V 0 is a Y -morphism, and g 0 φ = ψg. For any K ∈ ob D+ (Y, Z/n), we have H q (f ∗ K(d)[2d])x¯ ∼ =

H q (Rf ! K)x¯ ∼ =

lim −→

Extq (j! Z/n(−d)[−2d], K),

lim −→

Extq (f!· k! Z/n, K).

(U,V,g)∈ob S ◦

(U,V,g)∈ob S ◦

So we have biregular spectral sequences E2pq =

lim −→

Extp (H −q (j! Z/n(−d)[−2d]), K) ⇒ H p+q (f ∗ K(d)[2d])x¯ ,

lim −→

Extp (R−q f! k! Z/n, K) ⇒ H p+q (Rf ! K)x¯ ,

(U,V,g)∈ob S ◦

E2pq =

(U,V,g)∈ob S ◦

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and the morphism of complexes f!· k! Z/n → j! Z/n(−d)[−2d] introduced at the beginning of this section defines a morphism from the first spectral sequence to the second one. To prove the theorem, it suffices to show that the induced homomorphism lim −→

Extp (H

lim −→

Extp (R−q f! k! Z/n, K)

−q

(j! Z/n(−d)[−2d]), K)

(U,V,g)∈ob S ◦

→

(U,V,g)∈ob S ◦

is an isomorphism for any pair p, q. Given (U, V, g) ∈ ob S , by 8.5.7, we can find a commutative diagram k00

U0 g00

k0

k

j0

j

→ U ×V V 0 → U → X & ↓ g0 g ↓ f ↓ V0

→ V → Y

such that U 0 and V 0 are etale neighborhoods of x ¯ and f (¯ x), respectively, the canonical morphisms Rq g!00 Z/n → Rq g!0 Z/n vanishes for any q < 2d, and the restriction of Trg0 : R2d g!0 Z/n(d) → Z/n to the image of R2d g!00 Z/n(d) → R2d g!0 Z/n(d) is an isomorphism. So we can find a morphism α : Z/n → R2d g!0 Z/n(d) which is a section of Trg0 such that its composite with Trg00 coincides with R2d g!00 Z/n(d) → R2d g!0 Z/n(d). Trg00

R2d g!00 Z/n(d) → Z/n ↓ k α

R2d g!0 Z/n(d) Z/n. Trg0

We have a commutative diagram Rq f! (kk 0 k 00 )! Z/n ∼ = (jj 0 )! Rq g!00 Z/n ↓0 0 ↓ (jj )! Rq g!0 Z/n ↓ ∼ Rq f! k! Z/n = j! Rq g! Z/n

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for any q < 2d, and a commutative diagram 0

(jj )! (Trg00 ) R2d f! (kk 0 k 00 )! Z/n ∼ → (jj 0 )! Z/n(−d) = (jj 0 )! R2d g!00 Z/n ↓ k

(jj 0 )! R2d g!0 Z/n

↓ R2d f! k! Z/n

∼ =

(jj 0 )! (α)

(jj 0 )! (Trg0 )

↓

j! (Trg )

j! R2d g! Z/n

→

(jj 0 )! Z/n(−d) ↓

j! Z/n(−d).

These facts imply that the homomorphism lim −→

Extp (H

lim −→

Extp (R−q f! k! Z/n, K)

−q

(j! Z/n(−d)[−2d]), K)

(U,V,g)∈ob S ◦

→

(U,V,g)∈ob S ◦

is an isomorphism for any p, q. Indeed, when q 6= −2d, both sides are 0. When q = −2d, we have a commutative diagram Extp ((jj 0 )! R2d g!00 Z/n, K) ↑ Extp ((jj 0 )! R2d g!0 Z/n, K) p

↑

2d

Ext (j! R g! Z/n, K)

(jj 0 )! (Trg00 )

←

(jj 0 )! (α)

(jj 0 )! (Trg0 )

Extp ((jj 0 )! Z/n(−d), K) k Extp ((jj 0 )! Z/n(−d), K)

j! (Trg )

p

←

p

↑

Ext (j! Z/n(−d), K).

If e ∈ Ext (j! Z/n(−d), K) is mapped to 0 in

Extp (j! R2d g! Z/n, K) ∼ = Extp (R2d f! k! Z/n, K),

the above commutative diagram shows that it is mapped to 0 in Extp ((jj 0 )! Z/n(−d), K). This proves the homomorphism lim −→

(U,V,g)∈ob S ◦

Extp (j! Z/n(−d), K) →

lim −→

Extp (R2d f! k! Z/n, K)

(U,V,g)∈ob S ◦

is injective. For any e0 ∈ Extp (j! R2d g! Z/n, K) ∼ = Extp (R2d f! k! Z/n, K), let e ∈ Extp ((jj 0 )! Z/n(−d), K) be the image of e0 under the composite Extp (j! R2d g! Z/n, K)

→

Extp ((jj 0 )! R2d g!0 Z/n, K)

→

Extp ((jj 0 )! Z/n(−d), K).

(jj 0 )! (α)

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Then e and e0 have the same image in ∼ Extp (R2d f! (kk 0 k 00 )! Z/n, K). Extp ((jj 0 )! R2d g 00 Z/n, K) = !

This proves that the homomorphism lim −→

(U,V,g)∈ob S ◦

Extp (j! Z/n(−d), K) →

lim −→

Extp (R2d f! k! Z/n, K)

(U,V,g)∈ob S ◦

is surjective.

Proof of 8.5.7. We use induction on d. Let y = f (x). If d = 0, then f is etale and separated. Let Yey¯ be the strict localization of Y at y¯. By 2.8.14, there exists a section ˜j : Yey¯ → X ×Y Yey¯ of the base change fYey¯ : X ×Y Yey¯ → Yey¯ of f that maps y¯ to x¯. By 1.10.9, we can find an etale neighborhood V of

y¯ in Y and a section j : V → X ×Y V of the base change fV : X ×Y V → V that maps y¯ to x ¯. By 2.3.9, j is an open and closed immersion. Taking U = V , this proves the lemma for the case d = 0. Suppose d = 1. By 7.7.1, f is locally acyclic relative to Z/n. So for any algebraic geometric point t → Yey¯, we have Z/n if q = 0, ex¯ × e t, Z/n) ∼ H q (X = Yy¯ 0 if q 6= 0.

We thus have

e × e t, Z/n) ∼ lim H q (U = Yy¯ −→ e U

Z/n 0

if q = 0, if q = 6 0,

e goes over the family of etale neighborhoods of x where U ¯ in X ×Y Yey¯. By 7.2.10, H q (Xt , Z/n) is finite for any q, where Xt = X ×Y t. So there e of x exists an etale neighborhood U ¯ in X ×Y Yey¯ such that the canonical homomorphisms e × e t, Z/n) H q (Xt , Z/n) → H q (U Yy¯

f0

f be a U e -morphism between etale neighborare 0 for q = 1, 2. Let W → W e hoods of x ¯ in U , and let KW f (resp. KW f 0 ) be the kernel of the canonical homomorphism f × e t, Z/n) (resp. H 0 (Xt , Z/n) → H 0 (W f 0 × e t, Z/n)). H 0 (Xt , Z/n) → H 0 (W Yy¯

Yy¯

0

We have KW f ⊂ KW f 0 . Since H (Xt , Z/n) is finite, there exists an etale f of x e such that for any etale neighborhood W f 0 of x neighborhood W ¯ in U ¯ in 0 f0 ∼ f W , we have KW limW H (W ×Yey¯ t, Z/n) = Z/n, this implies f = KW f 0 . As − → f0 that the composite Z/n → H 0 (Xt , Z/n) → H 0 (Xt , Z/n)/KW f

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f is an etale neighborhood of x e , the canonis an isomorphism. Since W ¯ in U ical homomorphisms f × e t, Z/n) H q (Xt , Z/n) → H q (W Yy¯

are 0 for q = 1, 2. By 8.3.2, the canonical homomorphisms f × e t, µn ) → Hcq (Xt , µn ) Hcq (W Yy¯

are 0 for q = 0, 1, and the restriction of

TrXt /t : Hc2 (Xt , µn ) → Z/n to the image of f × e t, µn ) → H 2 (Xt , µn ) Hc2 (W c Ss¯

is an isomorphism. Here we use the fact that for any etale k-morphism π : U 0 → U between smooth algebraic curves over a separably closed field k, the transposes of the canonical homomorphisms H q (U, Z/n) → H q (U 0 , Z/n) through the pairing in 8.3.2 are the canonical homomorphisms Hc2−q (U 0 , µn ) → Hc2−q (U, µn ). This follows from the commutativity of the following diagram Hc2−q (U 0 , µn ) × ExtqU 0 (µn , µn ) ok ↓ Hc2−q (U, π! µn ) × ExtqU (π! µn , π! µn ) k ↓ Trπ Hc2−q (U, π! µn ) × ExtqU (π! µn , µn ) Trπ ↓ ↑ Tr∗π q 2−q Hc (U, µn ) × ExtU (µn , µn )

→ Hc2 (U 0 , µn ) ok → Hc2 (U, π! µn ) ↓ Trπ → Hc2 (U, µn ) k 2 → Hc (U, µn )

TrU 0 /k

&

TrU/k

%

Z/n,

the fact that the composite φ : ExtqU 0 (µn , µn ) → ExtqU (π! µn , µn ) of Tr

ExtqU 0 (µn , µn ) → ExtqU (π! µn , π! µn ) →π ExtqU (π! µn , µn ) is an isomorphism, and the fact that the composite Tr∗

φ−1

ExtqU (µn , µn ) →π ExtqU (π! µn , µn ) → ExtqU 0 (µn , µn ) can be identified with the canonical homomorphism H q (U, Z/n) → H q (U 0 , Z/n).

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e of x¯ in X ×Y Yey¯, denote by F q the images For any etale neighborhood U e U of the canonical morphisms Rq (f˜|Ue )! µn → Rq f˜! µn ,

where f˜ : X ×Y Yey¯ → induction to show that F eq = 0 for q = 0, 1 and U

Yey¯ is the base change of f . We use noetherian e such that there exists an etale neighborhood U the restriction of Trf˜ : R2 f˜! µn → Z/n

to FUe2 is an isomorphism. Let S be the set of those closed subsets A of Yey¯ e of x such that for any etale neighborhood U ¯ in X ×Y Yey¯, either F 0 |A 6= 0, e U

or FUe1 |A 6= 0, or the restriction of Trf˜|A to FUe2 |A is not an isomorphism. If S is not empty, then there exists a minimal element A in S , and A must be irreducible. Let t be an algebraic geometric point lying above the generic point of A. By our previous discussion, there exists an etale f of x¯ in X ×Y Yey¯ such that H q (W f × e t, µn ) → H q (Xt , µn ) neighborhood W c c Yy¯ are 0 for q = 0, 1, and the restriction of TrXt /t : Hc2 (Xt , µn ) → Z/n to the f × e t, µn ) → Hc2 (Xt , µn ) is image of the canonical homomorphism Hc2 (W Yy¯ q an isomorphism. So we have (F f )t = 0 for q = 0, 1, and the restriction W of (Tr ˜)t : (R2 f˜! µn )t → Z/n to (F 2 )t is an isomorphism. Since F q f W

f

f W

2 2 (q = 0, 1), ker (Trf˜) ∩ FW f and coker (Trf˜) ∩ FW f are constructible by 7.8.1, q there exists a nonempty open subset O of A such that F f |O (q = 0, 1), W 2 2 (ker (Trf˜) ∩ F f )|O and (coker (Trf˜) ∩ F f )|O are locally constant. As t lies W W in O, all these locally constant sheaves are 0. By the minimality of A, there q f 0 of x exists an etale neighborhood W ¯ in X ×Y Yey¯ such that F f |A−O = 0 W0 2 for q = 0, 1 and the restriction of Trf˜|A−O to F f0 |A−O is an isomorphism. W q f 00 = W f× f0 Let W ⊂ F q ∩ F q for all q. So e W . We have F (X×Y Yy¯ )

f 00 W

f

e U

f W

f0 W

q 2 2 Ff |A (q = 0, 1), (ker (Trf˜) ∩ FW f 00 )|A and (coker (Trf˜) ∩ FW f 00 )|A are all W 00 0. This contradicts A ∈ S . So S is empty. Therefore there exists an etale e of x neighborhood U ¯ in X ×Y Yey¯ such that F eq = 0 for q = 0, 1 and the U restriction of Tr ˜ : R2 f˜! µn → Z/n to F 2 is an isomorphism. By 5.9.8,

there exists an etale neighborhood V of y¯ in Y and an etale neighborhood e∼ U of x ¯ in X ×Y V such that U = U ×V Yey¯ and such that the assertions in 8.5.7 hold. This proves 8.5.7 in the case where d = 1. Suppose that 8.5.7 holds for smooth morphisms of relative dimensions < d, and let us prove that it holds for any smooth morphism f : X → Y pure of relative dimension d. Note that if we can find an etale neighborhood

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X 0 of x ¯ in X such that 8.5.7 holds for the morphism f |X 0 , then it holds for f . So to prove our assertion, we may assume there exists an etale Y -morphism X → AnY = Spec OY [t1 , . . . , tn ]. Let Y 0 = A1Y = Spec OY [t1 ]. We can factorize f as a composite h

g

X →Y0 →Y such that h and g are smooth pure of relative dimensions d0 = d − 1 and 1, respectively. By the induction hypothesis, we can find a commutative diagram U h00

→ X ×Y 0 V → X & ↓ h0 ↓h V → Y0

such that V is an etale neighborhood of h(¯ x) in Y 0 , U is an etale neighborhood of x ¯ in X ×Y 0 V , the canonical morphisms Rq h00! Z/n → Rq h0! Z/n are 0 for q 6= 2d0 , and the restriction of 0

Trh0 : R2d h0! Z/n(d0 ) → Z/n to the image of the canonical morphism 0

0

R2d h00! Z/n(d0 ) → R2d h0! Z/n(d0 ) is an isomorphism. Let g1 : V → Y be the composite of V → Y 0 and g : Y 0 → Y . Applying the d = 1 case to g1 , we get a commutative diagram j

Z g100

→ V ×Y W → V & ↓ g10 ↓ g1 W →Y

such that W is an etale neighborhood of y¯ in Y , Z is an etale neighborhood of h0 (¯ x) in V ×Y W , the canonical morphisms 00 0 Rq g1! Z/n → Rq g1! Z/n

are 0 for q 6= 2, and the restriction of 0 Trg10 : R2 g1! Z/n(1) → Z/n

to the image of the canonical morphism 00 0 R2 g1! Z/n(1) → R2 g1! Z/n(1)

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is an isomorphism. Consider the following commutative diagram: U

→ X ×Y 0 V % % ↓ h0 U ×Y W → X ×Y 0 V ×Y W V % % ↓ h1 % ↓ g1 → X ×Y 0 V ×V Z V ×Y W Y

U ×V Z

j

h02

&

% → 00

↓ h2 Z

↓ g10 W

g1

%

,

where h1 and h2 are the base changes of h0 , and h02 is the base change of h00 . When q 6= 2d0 , the canonical morphisms 00 q 0 0 Rq h1! Z/n(d0 ) Rp g1! R h2! Z/n(d0 ) → Rp g1!

are 0 since they are the composites of the canonical morphisms 00 q 0 00 q 0 Rq h1! Z/n(d0 ) R h2! Z/n(d0 ) → Rp g1! Rp g1! R h2! Z/n(d0 ) → Rp g1!

and the canonical morphisms Rq h02! Z/n(d0 ) → Rq h2! Z/n(d0 ) are 0. When p 6= 2, the canonical morphisms 0

0

00 2d 0 0 Rp g1! R h2! Z/n(d0 ) → Rp g1! R2d h1! Z/n(d0 )

are 0. To prove this, let h01 : U ×Y W → V ×Y W be the base change of h00 . The restrictions of Trhi (i = 1, 2) to the images of the canonical morphisms 0

0

R2d h0i! Z/n(d0 ) → R2d hi! Z/n(d0 ) are isomorphisms. Denote the inverses of these isomorphisms by Trh−1 . i Then these canonical morphisms can be factorized as Trh0

0

Tr−1 h

0

R2d h0i! Z/n(d0 ) →i Z/n →i R2d hi! Z/n(d0 ). The following diagram commutes: 0

00 2d 0 Rp g1! R h2! Z/n(d0 ) ok 0

Trh0

→2

Trh0

00 Rp g1! Z/n ok

Tr−1 h

→2

Tr−1 h

0

00 2d Rp g1! R h2! Z/n(d0 ) ok 0

0 0 0 Rp g1! j! j ∗ R2d h01! Z/n(d0 ) →1 Rp g1! j! j ∗ Z/n →1 Rp g1! j! j ∗ R2d h1! Z/n(d0 ) Trj ↓ Trj ↓ Trj ↓ 0

0 Rp g1! R2d h01! Z/n(d0 )

Trh0

→1

0 Rp g1! Z/n

Tr−1 h

→1

0

0 Rp g1! R2d h1! Z/n(d0 ).

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00 0 When p 6= 2, the canonical morphisms Rp g1! Z/n → Rp g1! Z/n are 0. So p 00 2d0 0 0 p 0 2d0 the canonical morphisms R g1! R h2! Z/n(d ) → R g1! R h1! Z/n(d0 ) are 0. Hence when p + q 6= 2d, the canonical morphisms 00 q 0 0 Rp g1! R h2! Z/n(d0 ) → Rp g1! Rq h1! Z/n(d0 )

are 0. We have biregular spectral sequences 00 q 0 E2pq = Rp g1! R h2! Z/n(d0 ) ⇒ Rp+q (g100 h02 )! Z/n(d0 ), 0 E2pq = Rp g1! Rq h1! Z/n(d0 ) ⇒ Rp+q (g10 h1 )! Z/n(d0 ).

It follows that the canonical morphisms Rq (g100 h02 )! Z/n(d0 ) → Rq (g10 h1 )! Z/n(d0 ) are 0 when q 6= 2d. φ ψ Let F → G → H be morphisms of sheaves. One can verify that the following three statements are equivalent: (a) ψ induces an isomorphism from im φ to H . (b) ψφ is surjective and ker (ψφ) ⊂ ker φ. (c) ψφ is surjective and there exists a morphism θ : H → G such that ψθ = idH and φ = θψφ. If these conditions holds, then for any right exact additive functor F , F (ψ) induces an isomorphism from im (F (φ)) to F (H ). Since the restriction of 0

Trh2 : R2d h2! Z/n(d0 ) → Z/n to the image of the canonical morphism 0

0

R2d h02! Z/n(d0 ) → R2d h2! Z/n(d0 ) 00 is an isomorphism and the functor R2 g1! is right exact, the restriction of the morphism 0

00 2d 00 R2 g1! R h2! Z/n(d0 ) → R2 g1! Z/n

induced by Trh2 to the image of the canonical morphism 0

0

00 2d 0 00 2d R2 g1! R h2! Z/n(d0 ) → R2 g1! R h2! Z/n(d0 )

is an isomorphism. Moreover, the restriction of 0 Trg10 : R2 g1! Z/n(1) → Z/n

to the image of the canonical morphism 00 0 R2 g1! Z/n(1) → R2 g1! Z/n(1)

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is an isomorphism. Using these facts and the following commutative diagram, 0

00 2d R2 g1! R h02! Z/n(d)

0

→

00 2d R2 g1! R h2! Z/n(d)

ok

ok

0 0 R2 g1! j! j ∗ R2d h01! Z/n(d)

→

0 0 R2 g1! j! j ∗ R2d h1! Z/n(d)

Trj ↓ 0

0 R2 g1! R2d h01! Z/n(d)

Trh 2

→

ok Trh 1

→

Trj ↓

→

00 R2 g1! Z/n(1)

Trj ↓

0

0 R2 g1! R2d h1! Z/n(d)

Trh 1

→

Tr 00 & g 1

0 R2 g1! j! j ∗ Z/n(1)

Z/n, Tr 0 % g1

0 R2 g1! Z/n(1)

one can check that the morphism 00 2d0 0 R2 g1! R h2! Z/n(d) → Z/n is surjective, and its kernel is contained in the kernel of the morphism 0 0 00 2d0 0 R2d h1! Z/n(d). R2 g1! R h2! Z/n(d) → R2 g1! So the restriction of 0 0 R2 g1! R2d h1! Z/n(d) → Z/n 0 00 2d0 0 0 to the image of R2 g1! R h2! Z/n(d) → R2 g1! R2d h1! Z/n(d) is an isomorphism. We have 00 2d0 0 R2d (g100 h02 )! Z/n(d) ∼ R h2! Z/n(d), = R2 g1! 0 2d 0 2 0 ∼ R (g h1 )! Z/n(d) = R g R2d h1! Z/n(d). 1

1!

It follows that the restriction of Trg10 h1 : R2d (g10 h1 )! Z/n(d) → Z/n to the image of the canonical morphism R2d (g100 h02 )! Z/n(d) → R2d (g10 h1 )! Z/n(d) is an isomorphism. Recall that we have shown the canonical morphisms Rq (g100 h02 )! Z/n(d) → Rq (g10 h1 )! Z/n(d) are 0 for q 6= 2d. Let gW : Y 0 ×Y W → W and hW : X ×Y W → Y ×Y W be the base changes of g and h, respectively. The above facts imply that the canonical morphisms Rq (g100 h02 )! Z/n(d) → Rq (gW hW )! Z/n(d) are 0 for any q 6= 2d, and the restriction of TrgW hW : R2d (gW hW )! Z/n(d) → Z/n to the image of the canonical morphism R2d (g100 h02 )! Z/n(d) → R2d (gW hW )! Z/n(d) is an isomorphism. This finishes the proof of the lemma. U ×V Z → X ×Y 0 V ×Y W → X ×Y W h02 ↓ h1 ↓ ↓ hW 0 Z → V ×Y W → Y ×Y W g100

&

↓ W

g10

gW

.

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Cohomology Classes of Algebraic Cycles

([SGA 4 21 ] Cycle.) Throughout this section, we fix a noetherian ring A such that A is an injective A-module, and nA = 0 for some integer n invertible on schemes we consider. Let S be a noetherian scheme, and let f : Y → Z be a smooth S-compactifiable morphism pure of relative dimension d between S-schemes of finite type. We have the trace morphism Trf : R2d f! A(d) → A.

On the other hand, we have Rq f! A(d) = 0 for all q > 2d and Rq f ! A(−d) = 0 for all q < −2d. So we have HomA (R2d f! A(d), A) ∼ = HomD(Z,A) (Rf! A(d), A[−2d]) ∼ = HomD(Y,A) (A(d), Rf ! A[−2d]) ∼ H 0 (Y, Rf ! A(−d)[−2d]) = ∼ = H 0 (Y, R−2d f ! A(−d)).

Hence Trf :R2d f! A(d)→ A corresponds to an element in H 0 (Y, Rf ! A(−d) [−2d]). Suppose that we have a smooth S-compactifiable morphism g : X → Z pure of relative dimension N and an immersion i : Y → X such that gi = f . Let c = N − d be the codimension of Y in X. We have Rf ! A ∼ = Ri! Rg ! A ∼ = Ri! A(N )[2N ].

So we have Rf ! A(−d)[−2d] ∼ = Ri! A(c)[2c]. It follows that Rq i! A = 0 for any q < 2c and H 0 (Y, Rf ! A(−d)[−2d]) ∼ = H 0 (Y, Ri! A(c)[2c]) ∼ = HY2c (X, A(c)). The element in HY2c (X, A(c)) corresponding to Trf is called the cohomology class associated to Y and is denoted by cl0 (Y ). The image of cl0 (Y ) under the canonical homomorphism HY2c (X, A(c)) → H 2c (X, A(c)) is also called the cohomology class of Y and is denoted by cl(Y ). The functors HYq (X, −) that we used above are defined as follows. Choose an open subset U of X containing i(Y ) as a closed subset. We define ΓY (X, F ) = ΓY (U, F |U )

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for any sheaf F on X. The functor ΓY (X, −) is independent of the choice of U and is left exact. Let RΓY (X, −) be the right derived functor of ΓY (X, −). We define HYq (X, F ) = H q (RΓY (X, F )) = HYq (U, F ).

Let k : Y → U and j : U → X be the immersions such that i = jk. We have ΓY (X, F ) = Γ(Y, k ! j ∗ F ). So we have RΓY (X, F ) = RΓ(Y, R(k ! j ∗ )F ) ∼ = RΓ(Y, Ri! F ). The next proposition shows that cl0 (Y ) ∈ HY2c (X, A(c)) depends only on i : Y → X and not on the base Z. Proposition 8.6.1. Let S be a noetherian scheme. Consider a commutative diagram X i

% ↓ g0 f0

Y → Z0 & ↓h f

Z, 0

where i is an immersion, f , f , g = hg 0 , g 0 and h are smooth Scompactifiable morphisms between S-schemes of finite type pure of relative dimensions d, d0 , N , N 0 and N − N 0 , respectively. Let c = N − d = N 0 − d0 .

The element in HY2c (X, A(c)) corresponding to Trf ∈ Hom(R2d f! A(d), A) 0 coincides with the element corresponding to Trf 0 ∈ Hom(R2d f!0 A(d0 ), A). Proof.

Use the commutativity of the following diagram:

Hom(Rf 0 A(d0 ), A[−2d0 ]) ! ∼ = ↓ Hom(A(d0 ), Rf 0! A[−2d0 ])

Trh →

(1)

Hom(Rh! Rf 0 A(d0 ), Rh! A[−2d0 ]) ! ∼ = ↓

→

Hom(A(d0 ), Rf 0! Rh! Rh! A[−2d0 ])

Trh →

→

∼ = ↓

∼ = ↓

Hom(Rf! A(d), A[−2d]) ∼ = ↓ Hom(A(d), Rf ! A[−2d]) ∼ = ↓

Trh Hom(A, Ri! Rg0! A(−d0 )[−2d0 ]) → Hom(A, Ri! Rg0! Rh! Rh! A(−d0 )[−2d0 ]) → Hom(A, Ri! Rg! A(−d)[−2d]) & ∼ =

(2) Hom(A, Ri! A(c)[2c]),

. ∼ =

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where (1) commutes because for any φ ∈ Hom(Rf!0 A(d0 ), A[−2d0 ]), the following diagram commutes: A(d0 ) → &

Rf 0! Rf!0 A(d0 ) ↓

Rf 0! (φ)

→

Rf 0! Rh! Rh! Rf!0 A(d0 )

Rf 0! Rh! Rh! (φ)

→

Rf 0! A[−2d0 ] ↓

Rf 0! Rh! Rh! A[−2d0 ],

and (2) commutes because A(−d0 )[−2d0 ] ∼ = Rh! A(−d)[−2d] and the following diagram commutes: Rh! A(−d)[−2d] = Rh! A(−d)[−2d] adj ↓ ↑ adj ! ! ! (Rh Rh! )Rh A(−d)[−2d] = Rh (Rh! Rh! )A(−d)[−2d]. Proposition 8.6.2. Let S be a noetherian scheme, g : X → Z and f : Y → Z smooth S-compactifiable morphisms between S-schemes of finite type pure of relative dimensions N and d, respectively, i : Y → X a closed immersion such that f = gi, and c = N − d. Then Trf : R2d f! A(d) → A is mapped to cl(Y ) ∈ H 2c (X, A(c)) under the composite of the following homomorphisms: Hom(R2d f! A(d), A) ∼ = Hom(Rf! A(d), A[−2d]) ∼ = Hom(Rg! i∗ i∗ A(d), A[−2d]) → Hom(Rg! A(d), A[−2d]) ∼ = Hom(A, Rg ! A(−d)[−2d]) ∼ = Hom(A, A(c)[2c]) ∼ = H 2c (X, A(c)).

Equivalently, through the isomorphisms Hom(Rg! A(d), A[−2d]) ∼ = Hom(A, Rg ! A(−d)[−2d]) ∼ = H 2c (X, A(c)), cl(Y ) ∈ H 2c (X, A(c)) corresponds to the composite Trf

Rg! A(d) → Rf! A(d) → A[−2d].

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Use the commutativity of the following diagram:

Proof.

Hom(Rf! A(d),A[−2d]) ∼ =↓

Hom(Rg! i∗ i∗ A(d),A[−2d])

=

Hom(Rg! i∗ i∗ A(d),A[−2d]) → Hom(Rg! A(d),A[−2d])

∼ =↓ ∗

Hom(i

∼ =↓

(1)

A(d),Ri Rg A[−2d]) ∼ = Hom(A(d),i∗ Ri! Rg! A[−2d]) !

!

∼ =↓

∼ =↓

→ Hom(A(d),Rg! A[−2d])

∼ =↓

∼ =

Hom(i∗ A,Ri! A(c)[2c])

∼ =↓

Hom(A,i∗ Ri! A(c)[2c])

→

Hom(A,A(c)[2c])

∼ =↓

∼ =↓

→

2c HY (X,A(c))

H

2c

(X,A(c)).

The commutativity of (1) follows from the adjointness of (Rg! , Rg ! ) and the following fact: For any sheaves F and G on X and any morphism ψ : i∗ i∗ F → G , if ψ 0 : F → i∗ i! G is the morphism induced by ψ by adjunction, then the composite adj

ψ

ψ0

adj

F → i∗ i∗ F → G coincides with the composite F → i∗ i! G → G . To prove this fact, note that ψ induces a morphism φ : i∗ F → i! G by adjunction such that ψ coincides with the composite i∗ φ

adj

i∗ i∗ F → i∗ i! G → G ,

and ψ 0 coincides with the composite adj

i∗ φ

F → i∗ i∗ F → i∗ i! G . Our assertion follows.

Let X be a smooth compactifiable scheme pure of dimension N over an algebraically closed field k, let Y be an integral closed subscheme of X pure of dimension d, and let c = N − d. There exists an open subset U of Y smooth over k. We can define cl0 (U ) ∈ HU2c (X, A(c)) as above. We claim that H 2c (X, A(c)) ∼ = H 2c (X, A(c)). U

Y

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We call the element in HY2c (X, A(c)) corresponding to cl0 (U ) the cohomology class associated to Y and denote it by cl0 (Y ). Let us prove our claim. Let Z = Y −U , and fix notation by the following commutative diagram

j

Z ↓l&h i

U → Y →

X,

where h, i, j, l are all immersions. We have a long exact sequence · · · → HZq (Y, Ri! A(c)) → H q (Y, Ri! A(c)) → H q (U, j ∗ Ri! A(c)) → · · · . Moreover, we have HZq (Y, Ri! A(c)) ∼ = H q (Z, Rl! Ri! A(c)) ∼ = H q (Z, Rh! A(c)), q H q (Y, Ri! A(c)) ∼ = H (X, A(c)), Y

q H q (U, j ∗ Ri! A(c)) ∼ = HU (X, A(c)).

To prove our claim, it suffices to show Rq h! A(c) = 0 for any q < 2(c + 1). Let g : X → Spec k be the structure morphism. We have Rh! A(c) ∼ = Rh! Rg ! A(c − N )[−2N ] ∼ = R(gh)! A(c − N )[−2N ]. Since dim Z ≤ d − 1, we have

Rq (gh)! A(c − N ) = 0

for any q < −2(d − 1). Hence

Rq h! A(c) ∼ = Rq−2N (gh)! A(c − N ) = 0

for any q − 2N < −2(d − 1), or equivalently, q < 2(c + 1). Next suppose that Y is a closed subscheme of X pure of dimension d, and let Y1 , . . . , Ym be all the irreducible components of Y with the reduced closed subscheme structure. We can define cl0 (Yi ) ∈ HY2ci (X, A(c)) as above. Denote their images in HY2c (X, A(c)) also by cl0 (Yi ). Let ηi be the generic point of Yi . We define the cohomology class associated to Y to be cl0 (Y ) =

m X i=1

0

length(OY,ηi ) cl0 (Yi ) ∈ HY2c (X, A(c)).

The image of cl (Y ) in H 2c (X, A(c)) is also called the cohomology class associated to Y and is denoted by cl(Y ).

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P Finally, let γ = i ni Yi be an algebraic cycle of codimension c. We define the cohomology class associated to γ to be X cl(γ) = ni cl(Yi ) ∈ H 2c (X, A(c)). i

Lemma 8.6.3. Consider a commutative diagram i0

Y0 j0 ↓

X0 ↓j

→ i

Y

f

→ &

X ↓g Spec k,

where k is an algebraically closed field, X and Y are smooth compactifiable k-schemes pure of dimensions N and d, respectively, i is a closed immersion, j is etale, and the square in the diagram is Cartesian. Let c = N − d. Then the image of cl0 (Y ) under the canonical homomorphism HY2c (X, A(c)) → HY2c0 (X 0 , A(c))

is cl0 (Y 0 ).

Proof. Let f 0 = f j 0 and g 0 = gj. We use the commutativity of the following diagram: Hom(Rf! A(d),A[−2d]) ∼ =↓

→ Hom(Rf! j!0 j 0∗ A(d),A[−2d]) ∼ = (1)

Hom(Rf!0 A(d),A[−2d])

∼ =↓

∼ =↓

0! Hom(A,Rf ! A(−d)[−2d]) → Hom(j 0∗ A(d),j 0∗ Rf ! A[−2d]) ∼ = Hom(A,Rf A(−d)[−2d]) ∼ =↓ !

Hom(A,Ri A(c)[2c])

∼ =↓ 0∗

→

0∗

∼ =↓ !

Hom(j A,j Ri A(c)[2c])

∼ =

Hom(A,Ri0! A(c)[2c]).

The commutativity of (1) follows from the fact that for any φ ∈ Hom(Rf! A, A(−d)[−2d]), the following diagram commutes: adj

j 0∗ A 0∗

j (adj) 0∗

!

j Rf A(−d)[−2d]

j 0∗ Rf ! (φ)

←

0∗

!

↓

j Rf Rf! A

→ (2) j 0∗ Rf ! Rf! (adj)

←

(j 0∗ Rf ! Rf! j!0 )j 0∗ A k

j 0∗ Rf ! Rf! (j!0 j 0∗ A).

To prove that (2) commutes, we use the adjointness of (j!0 , j 0∗ ) to reduce it to the commutativity of the following diagram: adj

j!0 j 0∗ A → Rf ! Rf! j!0 j 0∗ A adj ↓ ↓ Rf ! Rf! (adj) adj

A → Rf ! Rf! A.

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Lemma 8.6.4. Let S be a noetherian scheme. Consider a commutative diagram of Cartesian squares i0

g0

i

g

Y 0 → X0 → Z0 h00 ↓ h0 ↓ h↓ Y → X → Z.

Suppose all the schemes in the diagram are S-schemes of finite type, f = gi and g are smooth S-compactifiable morphisms pure of relative dimensions d and N , respectively, and i is a closed immersion. Let c = N − d. Then the image of cl0 (Y ) under the canonical homomorphism HY2c (X, A(c)) → HY2c0 (X 0 , A(c)) is cl0 (Y 0 ). Proof.

Let f 0 = g 0 i0 . The following diagram commutes: ∼ =

Hom(Rf! A(d), A[−2d]) ↓

Hom(A, Rf ! A(−d)[−2d])

∼ =

Hom(A, Ri! Rg! A(−d)[−2d])

↓

↓

Hom(h∗ Rf! A(d), h∗ A[−2d]) ∼ = Hom(A, Rf ! Rh∗ h∗ A(−d)[−2d]) ∼ = Hom(A, Ri! Rg! Rh∗ h∗ A(−d)[−2d]) ∼ = ↓

∼ = ↓

∼ = ↓

0! 0! ∗ 0! ∗ 00 ∼ Hom(Rf!0 h00∗ A(d), h∗ A[−2d]) ∼ = Hom(A, Rh00 ∗ Rf h A(−d)[−2d]) = Hom(A, Rh∗ Ri Rg h A(−d)[−2d]) ∼ ∼ ∼ = ↓ = ↓ = ↓ Hom(Rf!0 A(d), A[−2d])

∼ =

Hom(A, Rf 0! A(−d)[−2d])

∼ =

Hom(A, Ri0! Rg0! A(−d)[−2d]).

To prove our assertion, we need to show that the composite of the rightmost vertical arrows can be identified with the canonical homomorphism HY2c (X, A(c)) → HY2c0 (X 0 , A(c)). It suffices to show that the following diagram commutes: Ri! Rg ! A(−d)[−2d] ↓ ! ! Ri Rg Rh∗ h∗ A(−d)[2d] ok Ri! Rh0∗ Rg 0! h∗ A(−d)[−2d] ↓ 00 0! 0! ∗ Rh∗ Ri Rg h A(−d)[−2d]

& (1) Ri! Rh0∗ h0∗ Rg ! A(−d)[−2d] % ↓ Rh00∗ Ri0! h0∗ Rg ! A(−d)[−2d], %

where the slant arrows in the parallelogram in the lower part of the diagram is induced by the composite of the following isomorphisms t−1

tg g0 Rg 0! h∗ → g 0∗ h∗ (N )[2N ] ∼ = h0∗ g ∗ (N )[2N ] → h0∗ Rg ! .

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Here ∼ =

∼ =

tg : g ∗ (N )[2N ] → Rg ! ,

tg0 : g 0∗ (N )[2N ] → Rg 0!

are the isomorphisms in 8.5.2. The difficulty is to prove that the diagram (1) commutes. We will show in a moment that for any K ∈ ob D+ (Z 0 , A), the following diagram commutes: tg

g ∗ Rh∗ K(N )[2N ] ∼ =

↓

Rh0∗ g 0∗ K(N )[2N ]

Rg ! Rh∗ K

→ ∼ =

Rh0∗ (tg0 )

→ ∼ =

↑∼ =

(2)

Rh0∗ Rg 0! K,

where the vertical arrows are the canonical isomorphisms in 7.7.2 and 8.4.9, respectively. Thus through the isomorphisms tg and tg0 , the canonical morphism g ∗ Rh∗ (N )[2N ] → Rh0∗ g 0∗ (N )[2N ] can be identified with the inverse of the canonical morphism ∼ =

Rh0∗ Rg 0! → Rg ! Rh∗ . The commutativity of the diagram (1) then follows from the commutativity of the following diagram: g∗ → Rh0∗ h0∗ g ∗ ↓ ok ∗ ∗ 0 0∗ ∗ g Rh∗ h → Rh∗ g h . Let us prove that the diagram (2) commutes. By the adjointness of the functors (g ∗ , Rg∗ ), it suffices to show that the following diagram commutes: Rh∗ K(N )[2N ] ↓

Rh∗ Rg∗0 g 0∗ K(N )[2N ] ok

Rg∗ Rh0∗ g 0∗ K(N )[2N ]

&

Rh∗ Rg∗0 (tg0 )

→

Rg∗ Rh0∗ (tg0 )

→

→

Rg∗ Rg ! Rh∗ K

Rh∗ Rg∗0 Rg 0! K ok

k ∼ =

Rg∗ Rh0∗ Rg 0! K → Rg∗ Rg ! Rh∗ K,

where the horizontal arrow on the top of the diagram is obtained from tg : g ∗ Rh∗ K(N )[2N ] → Rg ! Rh∗ K through the adjoint functors (g ∗ , Rg∗ ), and the slant arrow is induced by the morphism K(N )[2N ] → Rg∗0 Rg 0! K

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obtained from tg0 through the adjoint functors (g 0∗ , Rg∗0 ). It is clear that the triangle and the square in this diagram commute. We are thus reduced to show the right part of the above diagram, that is, the following diagram, commutes: Rh∗ K(N )[2N ] → Rg∗ Rg ! Rh∗ K ↓ ↑∼ (3) = 0 0! ∼ Rh∗ Rg∗ Rg K = Rg∗ Rh0∗ Rg 0! K. Let k U →X g1 ↓ ↓g j

V →Z be a Cartesian diagram such that j is etale. Taking its base change with respect to h : Z 0 → Z, and denote the resulting Cartesian diagram by k0

U0 → X0 0 g1 ↓ ↓ g0 j0

V 0 → Z 0. Represent K by a bounded below complex of injective sheaves. By the definition of the isomorphisms tg and tg0 at the beginning of 8.5, to prove that the diagram (3) commutes, it suffices to show that the following diagram commutes: Hom(j! Z/n(−N )[−2N ], h∗ K) → Hom(g!· k! Z/n, h∗ K) (4) ok ↑ · Hom(j!0 Z/n(−N )[−2N ], K) → Hom(g 0 ! k!0 Z/n, K), where the first horizontal arrow is induced by the composite g!· k! Z/n → H 2N (g!· k! Z/n)[−2N ] ∼ = R2N g! k! Z/n[−2N ] ∼ =

j! (Trg1 )

j! R2N g1! Z/n[−2N ]

→ j! Z/n(−N )[−2N ], the second horizontal arrow is induced by the composite defined in the same way with g, k, j, g1 replaced by g 0 , k 0 , j 0 , g10 , respectively, and the right vertical arrow is induced by the canonical morphism · h∗ g!· k! Z/n → g 0 ! k!0 Z/n. The commutativity of the diagram (4) follows from 8.2.4 (i) applied to the Cartesian diagram U0 → U 0 g1 ↓ ↓ g1 V 0 → V.

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Suppose that X is a smooth compactifiable scheme pure of dimension N over an algebraically closed field k. Any Weil divisor D on X defines an ∗ element in H 1 (X, OX ) represented by the invertible OX -module L (D) by et 7.1.4 and 7.1.5. Kummer’s theory defines a homomorphism ∗ H 1 (X, OX ) → H 2 (X, µn ). et

Let cl(L (D)) ∈ H 2 (X, µn ) be the image of L (D) under this homomorphism. Proposition 8.6.5. Under the above assumption, we have cl(D) = cl(L (D)) in H 2 (X, µn ). Proof. It suffices to treat the case where D = Y is an integral closed subscheme of X of codimension 1. Let K ∗ be the etale sheaf on X so that K ∗ (U ) is the multiplicative group of the function field of U for any ∗ etale X-scheme U , and let O ∗ be the etale sheaf OX . The Weil divisor D et defines a Cartier divisor, that is, a section fD ∈ Γ(X, K ∗ /O ∗ ). It is clear that fD lies in ΓY (X, K ∗ /O ∗ ). Let δ : ΓY (X, K ∗ /O ∗ ) → HY1 (X, O ∗ ) be the homomorphism on cohomology groups arising from the short exact sequence 0 → O ∗ → K ∗ → K ∗ /O ∗ → 0. Then the image of δ f1D ∈ HY1 (X, O ∗ ) under the canonical homomorphism HY1 (X, O ∗ ) → H 1 (X, O ∗ )

is L (D). Kummer’s theory defines a homomorphism HY1 (X, O ∗ ) → HY2 (X, µn ).

Let cl0 (L (D)) ∈ HY2 (X, µn ) be the image of δ f1D under this homomorphism. Then the image of cl0 (L (D)) under the canonical homomorphism HY2 (X, µn ) → H 2 (X, µn )

is cl(L (D)). To prove the proposition, it suffices to show cl0 (D) = cl0 (L (D)) in HY2 (X, µn ). Let U be an open subset of X so that U ∩ Y is nonempty and smooth. Taking U sufficiently small, we may assume that there exists an etale morphism U → AN k = Spec k[t1 , . . . , tN ]

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such that U ∩Y ∼ = U ⊗k[t1 ,...,tN ] k[t1 , . . . , tN −1 ]. On the other hand, we have HY2 (X, µn ) ∼ = HY2 ∩U (X, µn ) ∼ = HY2 ∩U (U, µn ). So we may replace X by U . We then have a Cartesian diagram Y ↓

→ X ↓ i

−1 AN → AN k , k

−1 where i : AN → AN k is the closed immersion corresponding to the coork N −1 dinate plane tN = 0, and X → AN as a Weil divisor k is etale. Regard Ak N of Ak through this closed immersion. One can verify that cl0 (L (D)) is the −1 image of cl0 (L (AN )) under the canonical homomorphism k 2 HA2N −1 (AN k , µn ) → HY (X, µn ). k

0

−1 By 8.6.3, cl (D) is also the image of cl0 (AN ) under this homomorphism. k −1 So we are reduced to the case where Y → X is AN → AN k . We have a k commutative diagram i

−1 AN → AN k k id& ↓π −1 AN , k N −1 where π : AN is the projection to the first N − 1 coordinates. By k → Ak −1 8.6.1, when we define cl0 (Y ), we may use AN as the base scheme instead k of Spec k. The above commutative diagram can be obtained by base change from the diagram

{0}

→ ∼ = &

A1k ↓ Spec k.

Using 8.6.4, we are reduced to the case where Y → X is the closed immersion {0} → A1k , and then to the case where Y → X is the closed immersion {0} → P1k . We have a long exact sequence 2 · · · → H 1 (A1k , µn ) → H{0} (P1k , µn ) → H 2 (P1k , µn ) → · · · .

One can show H 1 (A1k , µn ) = 0. (Use 7.7.4, or Kummer’s theory and 7.1.2– 5.) So 2 H{0} (P1k , µn ) → H 2 (P1k , µn )

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is injective. To prove the proposition, it suffices to show cl(D) = cl(L (D)). Since TrP1k /k : H 2 (P1k , µn ) → Z/n is an isomorphism, it suffices to show TrP1k /k (cl(D)) = TrP1k /k (cl(L (D))). We have TrP1k /k (cl(L (D))) = deg(L (D)) = 1. By 8.6.2, through the Poincar´e duality Hom(H 0 (P1k , Z/n), Z/n) ∼ = H 2 (P1k , µn ), cl(D) corresponds to the composite H 0 (P1k , Z/n) → H 0 ({0}, Z/n)

Tr{0}/k

→

Z/n.

On the other hand, the homomorphism φ : H 0 (P1k , Z/n) → Z/n corresponding to cl(D) ∈ H 2 (P1k , µn ) has the property φ(1) = TrP1k /k (cl(D)). Using these facts, one proves TrP1k /k (cl(D)) = 1. Lemma 8.6.6. Let X be a smooth proper scheme of dimension d over an algebraically closed field k. The perfect pairing H q (X, Z/n) × H 2d−q (X, Z/n(d)) → Z/n defined by 8.5.3 coincides with the pairing H q (X, Z/n) × H 2d−q (X, Z/n(d)) → Z/n,

(s, t) 7→ TrX/k (t ∪ s).

Proof. Let C · (Z/n) be the Godement resolution of Z/n on X, and let I · be a resolution of Z/n(d) by injective sheaves of Z/n-modules on X. The canonical quasi-isomorphism Z/n → C · (Z/n) induces a quasi-isomorphism H om· (C · (Z/n), I · ) → H om· (Z/n, I · ). By 6.4.9, the stalk of C · (Z/n) at each geometric point is homotopically equivalent to Z/n. So the above quasi-isomorphism induces a quasiisomorphism H om· (C · (Z/n), I · ) ⊗Z/n C · (Z/n) → H om· (Z/n, I · ) ⊗Z/n C · (Z/n),

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and the stalkwise homotopic equivalence Z/n → C · (Z/n) induces a quasiisomorphism K · ⊗Z/n Z/n → K · ⊗Z/n C · (Z/n) for any complex of sheaves K . One can verify that the following diagram commutes: Ev

H om· (C · (Z/n),I ·)⊗Z/n Z/n → H om· (C · (Z/n),I ·)⊗Z/n C · (Z/n) → I · ↓ k H om· (Z/n, I · ) ⊗Z/n Z/n

Ev

I ·,

→

where Ev are the evaluation morphisms. The evaluation morphism on the second line is an isomorphism. It follows that the evaluation morphism on the first line is a quasi-isomorphism since all other morphisms in the above diagram are quasi-isomorphisms. We can find a commutative diagram (up to homotopy) Ev

H om· (C · (Z/n), I · ) ⊗Z/n C · (Z/n) → I · ↓ H om· (Z/n, I · ) ⊗Z/n C · (Z/n) ↓ ok I · ⊗Z/n C · (Z/n) → J· I· ∼ = I · ⊗Z/n Z/n → such that J · is a complex of injective sheaves of Z/n-modules, and all arrows in the diagram are quasi-isomorphisms. We have · ∼ · Z/n(d) ∼ = Z/n ⊗L Z/n Z/n(d) = C (Z/n) ⊗Z/n I

in D(X, Z/n). The cup product RΓ(X, Z/n(d)) ⊗L Z/n RΓ(X, Z/n) → RΓ(X, Z/n(d)) is the composite of the following morphisms RΓ(X, Z/n(d)) ⊗L Z/n RΓ(X, Z/n)

· ∼ = Γ(X, I · ) ⊗L Z/n Γ(X, C (Z/n))

→ Γ(X, I · ) ⊗Z/n Γ(X, C · (Z/n))

→ Γ(X, I · ⊗Z/n C · (Z/n))

→ Γ(X, J · ) ∼ = RΓ(X, Z/n(d)).

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The pairing in 8.5.3 is the composite H q (X, Z/n) × H 2d−q (X, Z/n(d)) ∼ = H q Γ(X, C · (Z/n)) × H 2d−q Γ X, H om· (C · (Z/n), I · ) → H 2d Γ X, H om· (C · (Z/n), I · ) ⊗Z/n C · (Z/n) Ev → H 2d Γ(X, I · ) ∼ = H 2d (X, Z/n(d)) TrX/k

→ Z/n.

Our assertion follows from the commutativity of the following diagram: H 2d Γ(X, H om· (C · (Z/n), I · ) Ev 2d H 2d−q (Γ(X, H om· (C · (Z/n), I · ))) → → H (Γ(X, I · )) q · × H (Γ(X, C (Z/n))) ⊗Z/n C · (Z/n)) ∼ =↓ H

2d−q

↓∼ = ·

·

(Γ(X, H om (Z/n, I )))

q

·

× H (Γ(X, C (Z/n)))

→

H

2d

Γ(X, H om· (Z/n, I · ) ⊗Z/n C · (Z/n))

∼ =↓

↓

↓∼ =

H 2d−q (Γ(X, I · )) × H q (Γ(X, C · (Z/n)))

→ H 2d (Γ(X, I · ⊗Z/n C · (Z/n)))

→ H 2d (Γ(X, J · )).

Proposition 8.6.7 (Lefschetz Fixed Point Formula). Let X be a smooth projective curve over an algebraically closed field k, h : X → X a k-morphism, Γh : X → X ×k X the graph of h, ∆ the divisor of X ×k X defined by the diagonal morphism ∆ : X → X ×k X, and Γ∗h (∆) the pulling back of the divisor ∆. Then we have 2 X q=0

(−1)q Tr(h∗ , H q (X, Z/n)) ≡ deg(Γ∗h (∆)) mod n.

Note that by 7.2.9 (ii), H q (X, Z/n) are free Z/n-modules for all q so that we can talk about Tr(h∗ , H q (X, Z/n)). Proof. Fix a primitive n-th root of unity and use it to define isomorphisms µn ∼ = Z/n and Z/n(d) ∼ = Z/n for all d. The homomorphism TrX/k : H 2 (X, Z/n(1)) → Z/n can be identified with a homomorphism H 2 (X, Z/n) → Z/n which we still denote by TrX/k . Similarly we have a homomorphism TrX×k X/k : H 4 (X ×k X, Z/n) → Z/n.

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For any s, t ∈ H 2 (X, Z/n), we have TrX×k X/k (π1∗ s ∪ π2∗ t) = TrX/k (s)TrX/k (t) by 8.2.4 (iv), where π1 , π2 : X ×k X → X are the projections. By 8.5.3 and 8.6.6, we have perfect pairings H q (X, Z/n) × H 2−q (X, Z/n) → Z/n, (s, t) 7→ TrX/k (s ∪ t),

H q (X ×k X, Z/n) × H 4−q (X ×k X, Z/n) → Z/n, (s, t) 7→ TrX×k X/k (s ∪ t). By 7.4.11, we have isomorphisms M H u (X, Z/n) ⊗ H v (X, Z/n) → H q (X ×k X, Z/n), u+v=q

s ⊗ t 7→ π1∗ s ∪ π2∗ t.

For each u, choose a basis {eu1 , . . . , eubu } for the free Z/n-module H u (X, Z/n). Let {fu1 , . . . , fubu } be the basis of H 2−u (X, Z/n) so that TrX/k (euα ∪ fuβ ) = δαβ . Then {π1∗ euα ∪ π2∗ fuβ }u,α,β is a basis of H 2 (X ×k X, Z/n). We claim that X cl(∆) = π1∗ fvγ ∪ π2∗ evγ . v,γ

Indeed, by 8.6.2, through the isomorphism Hom(H 2 (X ×k X, Z/n), Z/n) ∼ = H 2 (X ×k X, Z/n) induced by the perfect pairing H 2 (X ×k X, Z/n) × H 2 (X ×k X, Z/n) → Z/n,

(s, t) 7→ TrX×k X/k (s ∪ t),

cl(∆) ∈ H 2 (X ×k X, Z/n) corresponds to the composite TrX/k

∆∗

H 2 (X ×k X, Z/n) → H 2 (X, Z/n) → Z/n. So for any s ∈ H 2 (X ×k X, Z/n), we have TrX×k X/k (cl(∆) ∪ s) = TrX/k (∆∗ (s)). In particular, we have TrX×k X/k cl(∆) ∪ (π1∗ euα ∪ π2∗ fuβ ) = TrX/k ∆∗ (π1∗ euα ∪ π2∗ fuβ )

= TrX/k (∆∗ π1∗ euα ∪ ∆∗ π2∗ fuβ ) = TrX/k (euα ∪ fuβ ) = δαβ .

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On the other hand, we have X TrX×k X/k π1∗ fvγ ∪ π2∗ evγ ∪ (π1∗ euα ∪ π2∗ fuβ ) v,γ

= TrX×k X/k

X v,γ

= TrX×k X/k

X v,γ

= TrX×k X/k

(−1)u(2−v+v) π1∗ euα ∪ π1∗ fvγ ∪ π2∗ evγ ∪ π2∗ fuβ

X v,γ

π1∗ fvγ ∪ π2∗ evγ ∪ π1∗ euα ∪ π2∗ fuβ

π1∗ (euα ∪ fvγ ) ∪ π2∗ (evγ ∪ fuβ ) .

When u 6= v, we have either u + 2 − v ≥ 3 or v + 2 − u ≥ 3. So we have either euα ∪ fvγ = 0 or evγ ∪ fuβ = 0. Hence X TrX×k X/k π1∗ fvγ ∪ π2∗ evγ ∪ (π1∗ euα ∪ π2∗ fuβ ) v,γ

= TrX×k X/k =

X γ

=

X

X

π1∗ (euα

γ

∪ fuγ ) ∪

π2∗ (euγ

∪ fuβ )

TrX/k (euα ∪ fuγ )TrX/k (euγ ∪ fuβ ) δαγ δβγ

γ

= δαβ . Therefore TrX×k X/k (cl(∆) ∪ (π1∗ euα ∪ π2∗ fuβ )) X = TrX×k X/k π1∗ fvγ ∪ π2∗ evγ ∪ (π1∗ euα ∪ π2∗ fuβ ) .

So we have cl(∆) =

P

v,γ

∗ ∗ v,γ π1 fvγ ∪ π2 evγ . This proves our claim. Write X h∗ (evγ ) = avγδ evδ , δ

where avγδ ∈ Z/n. Then we have

Γ∗h (cl(∆)) = Γ∗h =

X

X v,γ

=

X v,γ

=

X

v,γ

π1∗ fvγ ∪ π2∗ evγ

Γ∗h π1∗ fvγ ∪ Γ∗h π2∗ evγ fvγ ∪ h∗ evγ

v,γ,δ

avγδ fvγ ∪ evδ .

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So we have

X TrX/k Γ∗h (cl(∆)) = avγδ TrX/k (fvγ ∪ evδ ) v,γ,δ

=

X

v,γ,δ

=

X

(−1)v(2−v) avγδ TrX/k (evδ ∪ fvγ )

(−1)v avγδ δγδ

v,γ,δ

=

X

(−1)v avγγ

v,γ

=

X

(−1)v Tr(h∗ , H v (X, Z/n)).

v

On the other hand, we have

Γ∗h (cl(∆)) = Γ∗h cl(L (∆)) = cl L (Γ∗h (∆)) .

So

Therefore

TrX/k Γ∗h (cl(∆)) ≡ deg(Γ∗h (∆)) mod n.

X v

(−1)v Tr(h∗ , H v (X, Z/n)) ≡ deg(Γ∗h (∆)) mod n.

Corollary 8.6.8. Let X be a smooth projective curve over an algebraically closed field k and let h : X → X be a k-morphism with isolated fixed points x1 , . . . , xm . For each xi , let vxi be the valuation on the function field of X defined by xi , and let πxi ∈ OX,xi be a uniformizer for vxi . Then we have 2 X j=0

(−1)j Tr(h∗ , H j (X, Z/n)) ≡

m X i=1

vxi (πxi − h∗ (πxi )) mod n.

Remark 8.6.9. Note that the number vxi (πxi − h∗ (πxi )) is independent of the choice of the uniformizer πxi of vxi . In fact, it is the largest integer j so that h is the identity on OX,xi /mjxi . We call vxi (πxi − h∗ (πxi )) the multiplicity of the fixed point xi . Proof. Let x be a closed point of X. Let us calculate the multiplicity of the divisor Γ∗h (∆) at x. If x is not a fixed point of h, then (x, h(x)) does not lie in ∆, and hence the multiplicity of Γ∗h (∆) at x is 0. Suppose that x is one of the isolated fixed point xi . Since X is smooth, we can find an open neighborhood U of x in X admitting an etale morphism f : U → A1k

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that maps x to the origin of A1k . Let ∆U and ∆A1k be the diagonal divisors of U ×k U and A1k ×k A1k , respectively. Consider the commutative diagram ∆f

U → U ×A1k U → f & ↓ A1k

∆ A1

U ×k U ↓ f ×f

→k A1k ×k A1k .

The square in the diagram is Cartesian, and the diagonal morphism ∆f is an open immersion by 2.3.9. ∆U coincides with the composite ∆f

U → U ×A1k U → U ×k U. It follows that the divisors ∆U and (f ×f )∗ (∆A1k ) coincide in a neighborhood of (x, x) in U ×k U . Hence the divisors Γ∗h (∆) and (f, f h)−1 (∆A1k ) coincide in a neighborhood of x in X. The divisor ∆A1k on A1k ×k A1k ∼ = Spec (k[t]⊗k k[t]) is the principle divisor defined by t ⊗ 1 − 1 ⊗ t. So in a neighborhood of x in X, Γ∗h (∆) is the principle divisor defined by f ∗ (t) − h∗ f ∗ (t). Since f is etale, f ∗ (t) is a uniformizer for vx . For any uniformizer πx of vx , we have vx (f ∗ (t) − h∗ f ∗ (t)) = vx (πx − h∗ (πx )). So we have deg(Γ∗h (∆)) =

m X i=1

We then apply 8.6.7.

vxi (πxi − h∗ (πxi )).

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9.1

Sheaves with Group Actions

Let G be a group acting on a scheme X on the right. For any etale X-scheme p : V → X, and any σ ∈ G, let Vσ be the etale X-scheme σ −1 p : V → X, and let V ×X,σ X be the etale X-scheme obtained from p : V → X by the base change σ : X → X. Then the projection V ×X,σ X → Vσ is an isomorphism of etale X-schemes. Let F be a sheaf on X. An action of G on F compatible with the action of G on X is a family of isomorphisms φσ : F → σ∗ F

(σ ∈ G)

such that φe = id and for any σ, τ ∈ G, the following diagram commutes: φσ

F → σ∗ F φτ σ ↓ ↓ σ∗ (φτ ) (τ σ)∗ F ∼ = σ∗ (τ∗ F ). Equivalently, an action of G on F is a family of isomorphisms φσ,V : F (V ) → F (Vσ ) for any σ ∈ G and any etale X-scheme V such that for any X-morphism V 0 → V of etale X-schemes and any σ, τ ∈ G, the following diagrams commute: φσ,V

F (V ) → F (Vσ ) ↓ ↓ φσ,V 0

F (V 0 ) → F (Vσ0 ),

φσ,V

F (V ) → F (Vσ ) φτ σ,V ↓ ↓ φτ,Vσ F (Vτ σ ) = F ((Vσ )τ ).

We can also define an action of G on F to be a family of isomorphisms ψσ : σ ∗ F → F 497

(σ ∈ G)

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such that ψe = id and for any σ, τ ∈ G, the following diagram commutes: τ ∗ (σ ∗ F ) ok (τ σ)∗ F

τ ∗ (ψσ )

→

ψτ σ

→

τ ∗F ↓ ψτ F.

A sheaf F on X with an action of G is called a G-sheaf. We define morphisms of G-sheaves to be morphisms of sheaves commuting with the group actions. For all G-sheaves F and G on X, denote by HomG (F , G ) the set of G-morphisms from F to G . Let F be a sheaf on X. Define a G-sheaf F [G] by M F [G] = σ∗ F , σ∈G

where for any τ ∈ G, φτ : F [G] → τ∗ (F [G]) is the composite F [G] ∼ =

M

σ∈G

(στ )∗ F ∼ =

M

σ∈G

τ∗ (σ∗ F ) ∼ = τ∗ (F [G]).

For any G-sheaf G on X, we have a one-to-one correspondence Hom(F , G ) ∼ = HomG (F [G], G ) which is functorial in F and G . So the functor F → F [G] from the category of sheaves to the category of G-sheaves is left adjoint to the forgetful functor from the category of G-sheaves to the category of sheaves. As the functor F → F [G] is exact, if G is injective in the category of G-sheaves, then it is injective in the category of sheaves. For any etale X-scheme p : V → X, let ZV = p! Z. We have HomG (ZV [G], G ) ∼ = Hom(ZV , G ) ∼ = G (V ). It follows that ZV [G] form a family of generators for the category of Gsheaves. Using this fact, one proves that the category of G-sheaves has enough injective objects. (Confer [Grothendieck (1957)] 1.10.1, or [Fu (2006)] 2.1.6.) Let Y be another scheme on which G acts on the right, and let f : X → Y be a G-equivariant morphism. For any G-sheaf F on X and G-sheaf G on Y , f∗ F and f ∗ G are G-sheaves on Y and on X, respectively, and

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the canonical morphisms G → f∗ f ∗ G and f ∗ f∗ F → F are G-morphisms. This follows from the commutativity of the following diagrams: G → f∗ f ∗ G → f∗ σ∗ σ ∗ f ∗ G ∼ = f∗ σ∗ f ∗ σ ∗ G ↓ ↓ ∼ σ∗ G → σ∗ f∗ f ∗ G f∗ σ∗ f ∗ G , = ∼ f ∗ σ ∗ f∗ F σ ∗ f ∗ f∗ F → σ∗ F = ↓ ↓ f ∗ σ ∗ f∗ σ∗ F ∼ → f ∗ f∗ F → F . = f ∗ σ ∗ σ∗ f∗ F To prove the commutativity of these diagrams, note that they are the outer loops of the following diagrams: ∼ f∗ σ∗ f ∗ σ ∗ G G → f∗ f ∗ G → f∗ σ∗ σ ∗ f ∗ G = ↓ ↓ ↓ ↓ ∗ ∗ ∗ ∗ ∗ ∼ σ∗ G → f∗ f σ∗ G → f∗ σ∗ σ f σ∗ G = f∗ σ∗ f σ σ∗ G k (1) ↓ ∼ σ∗ G → σ∗ f∗ f ∗ G f∗ σ∗ f ∗ G , = ∼ f ∗ σ ∗ f∗ F σ ∗ f ∗ f∗ F → σ∗ F = ↓ (2) k ∼ f ∗ σ ∗ σ∗ f∗ σ ∗ F f ∗ σ ∗ f∗ σ∗ σ ∗ F = → f ∗ f∗ σ ∗ F → σ ∗ F ↓ ↓ ↓ ↓ f ∗ σ ∗ f∗ σ∗ F ∼ → f ∗ f∗ F → F . = f ∗ σ ∗ σ∗ f∗ F

The diagrams (1) and (2) commute by 7.3.16. Since f ∗ is an exact functor, f∗ maps injective G-sheaves on X to injective G-sheaves on Y . If f : X → Y is an etale G-equivariant morphism, and F a G-sheaf on X, then f! F is a G-sheaf on Y , and the canonical morphism F → f ∗ f! F is a morphism of G-sheaves. This follows from the commutativity of the following diagram: σ∗ F = σ∗ F → F ↓ ↓ ↓ . ∼ = ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ f f! F → f σ f! F → f f! σ F → f f! F .

For any G-sheaf G on Y , the canonical morphism f! f ∗ G → G is a morphism of G-sheaves. This follows from the commutativity of the following diagram: ∼ =

σ ∗ f! f ∗ G → f! σ ∗ f ∗ G → f! f ∗ σ ∗ G → f! f ∗ G ↓ ↓ ↓ σ∗ G = σ∗ G → G .

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f! is left adjoint to f ∗ . Since f! is an exact functor, f ∗ maps injective G-sheaves on Y to injective G-sheaves on X. Suppose G acts trivially on a scheme S. For any G-sheaf F on S, let F G be the subsheaf formed by G-invariant sections. Let G be a sheaf on S. Put the trivial G-action on G . Then we have HomG (G , F ) ∼ = Hom(G , F G ). So the functor that maps G to the G-sheaf G with the trivial G-action is left adjoint to the functor F 7→ F G , and it is exact. If F is injective in the category of G-sheaves, then F G is injective in the category of sheaves. We denote the q-th right derived functor of F 7→ F G by H q (G, −) or Rq ΓG . Let X be a scheme on which G acts on the right, let Y be a scheme with the trivial G-action, and let f : X → Y be a G-equivariant morphism. For any G-sheaf F on X, let f∗G F = (f∗ F )G . The functor f∗G is left exact. Since f∗ maps injective G-sheaves on X to injective G-sheaves on Y , we have a biregular spectral sequence E2pq = H p (G, Rq f∗ F ) ⇒ Rp+q f∗G F . Let S be a scheme with the trivial G-action, let g : Y → S be a morphism, and let h = gf . We have G hG ∗ F = g∗ f∗ F .

Since f∗G maps injective G-sheaves on X to injective sheaves on Y , we have a biregular spectral sequence E2pq = Rp g∗ Rq f∗G F ⇒ Rp+q hG ∗ F. Consider the case where f : X → Y is a galois etale covering with galois group G = Aut(X/Y )◦ . G acts on the right of X, and acts trivially on Y . For any sheaf G on Y , the canonical morphism G → f∗ f ∗ G induces an isomorphism G ∼ = f G f ∗G . ∗

To prove this, we choose a surjective etale morphism Y 0 → Y such that X ×Y Y 0 → Y 0 is a trivial etale covering space. We then use the fact that ∼ (f∗ f ∗ G )G |Y 0 . G |Y 0 = Let 0 → I0 → I1 → ···

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be an injective resolution of G . Then 0 → f ∗I 0 → f ∗ I 1 → · · · is an injective resolution of f ∗ G in the category of G-sheaves. So Rq f∗G (f ∗ G ) is the q-th cohomology sheaf of the complex 0 → f∗G f ∗ I 0 → f∗G f ∗ I 1 → · · · . But we have I · ∼ = f∗G f ∗ I . So we have G if q = 0, q G ∗ R f∗ (f G ) = 0 if q ≥ 1. Let g : Y → S be a morphism of schemes with trivial G-actions, and let h = gf . The biregular spectral sequence ∗ E2pq = Rp g∗ Rq f∗G (f ∗ G ) ⇒ Rp+q hG ∗ (f G ) degenerates, and we have ∗ R p g∗ G ∼ = R p hG ∗ (f G ). The biregular spectral sequence ∗ E2pq = H p (G, Rq h∗ (f ∗ G )) ⇒ Rp+q hG ∗ (f G ) can be identified with a biregular spectral sequence E2pq = H p (G, Rq h∗ (f ∗ G )) ⇒ Rp+q g∗ G , which is called the Hochschild–Serre spectral sequence. Let X be a scheme on which G acts on the right, and let A be a ring. We define Ext·A[G] (−, −) and E xt·A[G] (−, −) to be the derived functors of HomA[G] (−, −) and H omA[G] (−, −) on the category of G-sheaves of Amodules on X, respectively. Suppose that G acts trivially on X and F is a G-sheaf of A-modules on X. We have F G = H omA[G] (A, F ), where A is the constant sheaf on which G acts trivially. So we have H p (G, F ) = E xtpA[G] (A, F ). Let · · · → L−1 → L0 → 0 be a resolution of A by free A[G]-modules of finite rank. Then we have H p (G, F ) = H p (H omG (L· , F )). Using this fact, one can show that if F is locally constant, then H p (G, F ) are locally constant, and that if f : X 0 → X is a morphism of schemes and G acts trivially on X 0 , then f ∗ H p (G, F ) ∼ = H p (G, f ∗ F ). In particular, we have H p (G, F )x¯ ∼ = H p (G, Fx¯ ) for any x ∈ X.

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Nearby Cycle and Vanishing Cycle

([SGA 7] I 2, XIII.) Let k be a field, k¯ a separable closure of k, Y a scheme over k, and Y = ¯ Then Gal(k/k) ¯ ¯ Y ⊗k k. acts on Y . Let F be a Gal(k/k)-sheaf on Y . For 0 ¯ let Y 0 = Y ⊗k k 0 . Then any finite galois extension k of k contained in k, ¯ 0 ) acts on F (U 0 ×Y 0 Y ). If this for any etale morphism U 0 → Y 0 , Gal(k/k action is continuous with respect to the discrete topology on F (U 0 ×Y 0 Y ) for any finite extension k 0 of k contained in k¯ and any quasi-compact scheme ¯ U 0 etale over Y 0 , we say that Gal(k/k) acts continuously on F . To check ¯ ¯ acts Gal(k/k) acts continuously on F , it suffices to check that Gal(k/k) 0 0 0 continuously on F (U ×Y 0 Y ) for any etale Y -scheme U such that U 0 is affine and its image in Y 0 is contained in the inverse image in Y 0 of an affine open subset of Y . Proposition 9.2.1. Let k be a field, k¯ a separable closure of k, Y a scheme ¯ and F the inverse image of F on over k, F a sheaf on Y , Y = Y ⊗k k, ¯ Y . Then Gal(k/k) acts on F . ¯ (i) The action of Gal(k/k) on F is continuous. (ii) The functor F 7→ F is an equivalence from the category of sheaves ¯ on Y to the category of sheaves on Y with continuous Gal(k/k)-action. Proof. (i) Let k 0 be a finite extension of k, let Y 0 = Y ⊗k k 0 , and let U 0 be an affine etale Y 0 -scheme such that its image in Y 0 is contained in an affine open subset of Y 0 . We have k¯ = lim kλ , where kλ are finite extensions of k −→ ¯ For each λ, let Fλ be the inverse image containing k 0 and contained in k. of F on Y ⊗k kλ . By 5.9.3, we have F (U 0 ×Y 0 Y ) = lim Fλ (U 0 ⊗k0 kλ ). −→ λ

¯ λ ) acts trivially on Fλ (U 0 ⊗k0 kλ ). It follows that For each λ, Gal(k/k 0 ¯ Gal(k/k ) acts continuously on F (U 0 ×Y 0 Y ). (ii) We have k¯ = lim kλ , where kλ are finite galois extensions of k con−→ ¯ Let Yλ = Y ⊗k kλ , and let π : Y → Y , πλ : Yλ → Y and tained in k. π ¯λ : Y → Yλ and be the projections. Denote the functor F 7→ F by S, and define a functor T from the category of sheaves on Y with continuous ¯ Gal(k/k)-action to the category of sheaves on Y by ¯

T (F ) = (π∗ F )Gal(k/k) .

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To prove that S is an equivalence of categories, it suffices to show that the canonical morphisms ¯ ¯ ∗ Gal(k/k) ∗ Gal(k/k) F → (π∗ π F ) , π (π∗ F ) →F

are isomorphisms for any sheaf F on Y , and any sheaf F on Y with ¯ continuous Gal(k/k)-action. By 5.9.6, we have ¯ ∗ Gal(k/k) (π∗ π F ) = lim(πλ∗ πλ∗ F )Gal(kλ /k) . −→ Note that F → (πλ∗ πλ∗ F )Gal(kλ /k) is an isomorphism for each λ. This can be seen by taking the base change from k to kλ . It follows that F → ¯ (π∗ π ∗ F )Gal(k/k) is an isomorphism. ¯

To prove that the morphism π ∗ (π∗ F )Gal(k/k)

→ F is an isomor-

¯ it suffices to phism for any sheaf F on Y with continuous Gal(k/k)-action, prove that the above morphism is surjective for any F . Indeed, suppose that this can be done and let K be the kernel of the above morphism. Since T is left exact, we have an exact sequence 0 → T (K ) → T (ST (F )) → T (F ). We have shown that the canonical morphism T (F ) → (T S)T (F ) is an isomorphism. One can show that it is a right inverse of the canonical morphism T (ST (F )) → T (F ). It follows that T (K ) = 0. But the canonical morphism ST (K ) → K is surjective. So we have K = 0 and hence the canonical morphism ST (F ) → F is an isomorphism. First let us show that the canonical morphism ¯ λ) Gal(k/k ∗ →F lim π ¯ (¯ πλ∗ F ) −→ λ is surjective. Let U be an affine etale Y -scheme so that its image in Y is contained in the inverse image in Y of an affine open subset of Y . We can find λ0 and a quasi-compact etale Yλ0 -scheme Uλ0 such that U ∼ = Y ×Yλ0 Uλ0 . For any λ ≥ λ0 , let Uλ = Yλ ×Yλ0 Uλ0 . Given s ∈ F (U ), ¯ λ ). Then s lies in the there exists λ1 ≥ λ0 such that s is fixed by Gal(k/k 1 image of ¯

π ¯λ∗1 (¯ πλ1 ∗ F )Gal(k/kλ1 ) (U ) → F (U ).

This proves our assertion. We have ¯ ¯ Gal(k/k) ∗ ∼ ¯λ∗ πλ∗ (πλ∗ π ¯λ∗ F )Gal(k/k) π (π∗ F ) =π ∼ ¯λ∗ πλ∗ =π

Gal(kλ /k) ! ¯ λ) Gal(k/k . πλ∗ (¯ πλ∗ F )

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For any sheaf Fλ on Yλ provided with a Gal(kλ /k)-action, the canonical morphism ∗ Gal(kλ /k) πλ (πλ∗ Fλ ) → Fλ is an isomorphism. This can be seen by taking the base change from k to kλ . So we have ¯ ¯ λ) ∗ Gal(k/k π ¯ (¯ π F ) . π ∗ (π∗ F )Gal(k/k) ∼ = λ λ∗

Since

lim π ¯∗ −→ λ

¯ λ) ¯ Gal(k/k ∗ Gal(k/k) (¯ πλ∗ F ) → F is surjective, π (π∗ F ) →

F is also surjective.

Let S be a trait, s its closed point, η its generic point, Se the strict e We have an exact localization of S at s¯, and η˜ the generic point of S. sequence 0 → I → Gal(¯ η /η) → Gal(¯ s/s) → 0,

where I = Gal(¯ η /˜ η) is the inertia subgroup. We have a commutative diagram η¯ → S˜ ← s¯ ↓ ↓ ↓ η → S ← s.

For any S-scheme X, taking the base change X → S to the above diagram, we get ˜

˜i j e = X ×S S˜ ← Xη¯ → X Xs¯ ↓ ↓ ↓ Xη → X ← Xs .

For any sheaf Fη on Xη , denote its inverse image in Xη¯ by Fη¯. By 9.2.1, Fη is completely determined by the sheaf Fη¯ with Gal(¯ η /η)-action. The group Gal(¯ η /η) acts on the scheme Xs¯ though its quotient group Gal(¯ s/s). ∗˜ ˜ η /η)-sheaf on Xs¯. Indeed, given σ ∈ Gal(¯ η /η), The sheaf i j∗ Fη¯ is a Gal(¯ ˜ it gives rise to an element in Aut(S/S) and an element in Gal(¯ s/s). So it ˜ gives rise to elements in Aut(Xη¯/Xη ), Aut(X/X), and Aut(Xs¯/Xs ), which ∗ we still denote by σ. The action of σ on ˜i ˜j∗ Fη¯ is given by the isomorphism σ ∗˜i∗˜j∗ Fη¯ → ˜i∗˜j∗ Fη¯ defined by the composite σ ∗˜i∗˜j∗ Fη¯ ∼ = ˜i∗ σ ∗ ˜j∗ Fη¯ → ˜i∗ ˜j∗ σ ∗ Fη¯ → ˜i∗˜j∗ Fη¯ .

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Define Ψη (Fη ) = ˜i∗ ˜j∗ Fη¯ . Ψη is a functor from the category of sheaves on Xη to the category of sheaves on Xs¯ with Gal(¯ η /η)-action. Its derived functor RΨη is called the nearby cycle functor. We have RΨη (Kη ) = ˜i∗ R˜j∗ Kη¯ for any Kη ∈ ob D+ (Xη ). Proposition 9.2.2. Keep the above notation. ex¯ be the strict localization of X at x (i) For any x ∈ Xs¯, let X ¯. It is a e We have (Rq Ψη Fη )x¯ ∼ ex¯ × e η¯, Fη¯ ). scheme over S. = H q (X S (ii) If f : X → S is of finite type, then RΨη has finite cohomological + dimension on Dtor (Xη ). More precisely, we have Rq Ψη (Fη ) = 0 for any q > dim Xη and any torsion sheaf Fη on Xη . (iii) If X = S, then we have RΨη (Kη ) = Kη¯, where on the right-hand side, Kη¯ is considered as a complex of sheaves on s¯ with Gal(¯ η /η) action. Proof. (i) We have ex¯ × e η¯, Fη¯). (Rq Ψη Fη )x¯ = (˜i∗ Rq ˜j∗ Fη¯ )x¯ ∼ = (Rq ˜j∗ Fη¯ )x¯ ∼ = H q (X S

(ii) Use 7.5.7. (iii) This has been noted in the proof of 7.6.1. Let i : s¯ → S˜ and j : η¯ → S˜ be the canonical morphisms. We have Rj∗ Kη¯ = j∗ Kη¯, and i∗ j∗ Kη¯ = Kη¯. Let F be a sheaf on X. Denote its inverse images on Xs , Xs¯, Xη , Xη¯, f, respectively. By 9.2.1, Fs is completely e by Fs , Fs¯, Fη , Fη¯ , and F and X determined by Fs¯ with Gal(¯ s/s)-action. Let Gal(¯ η /η) act on Fs¯ though its quotient Gal(¯ s/s). Then the canonical morphism f = ˜i∗ ˜j∗ Fη¯ f → ˜i∗˜j∗ ˜j ∗ F Fs¯ = ˜i∗ F

is a morphism of Gal(¯ η /η)-sheaves on Xs¯. If F is an injective sheaf, then Rq Ψη (Fη ) = 0 for all q > 0. This follows from 5.9.9 and the fact that Xη¯ is the inverse limit of an inverse system of etale X-schemes. e be its For any complex of sheaves K on X, let Ks , Ks¯, Kη , Kη¯, and K e respectively. Suppose that K inverse images on Xs , Xs¯, Xη , Xη¯, and X

is bounded below and let K → J be a quasi-isomorphism such that J is a bounded below complex of injective sheaves. We have e RΨη (Kη ) ∼ = ˜i∗˜j∗ ˜j ∗ J.

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We have a canonical morphism of complexes of Gal(¯ η /η)-sheaves on Xs¯: e Js¯ → ˜i∗˜j∗ ˜j ∗ J.

Let RΦ(K) be its mapping cone. Then RΦ defines a functor from D+ (X) to the derived category of Gal(¯ η /η)-sheaves on Xs¯. It is called the vanishing cycle functor. We have a distinguished triangle Ks¯ → RΨη (Kη ) → RΦ(K) → for any K ∈ ob D+ (X). Suppose that σ ∈ Gal(¯ η /η) lies in the inertia subgroup I. Then σ acts trivially on Ks¯. There exists a morphism var(σ) : RΦ(K) → RΨη (Kη ), which we call the variation, such that the following diagram commute: σ−id

RΨη (Kη ) → RΨη (Kη ) ↓ % var(σ) ↓ RΦ(K)

σ−id

→

RΦ(K).

For each q, the composite of the canonical homomorphisms e R˜j∗ Kη¯) → H q (Xs¯, ˜i∗ R˜j∗ Kη¯) = H q (Xs¯, RΨη (Kη )) H q (Xη¯, Kη¯) ∼ = H q (X,

defines a homomorphism

H q (Xη¯ , Kη¯) → H q (Xs¯, RΨη (Kη )), + which is an isomorphism if X is proper over S and K ∈ ob Dtor (X) by the proper base change theorem 7.3.3. We have a long exact sequence

· · · → H q (Xs¯, Ks¯) → H q (Xs¯, RΨη (Kη )) → H q (Xs¯, RΦ(K)) → · · · . + If X is proper over S and K ∈ ob Dtor (X), we then have a long exact sequence

· · · → H q (Xs¯, Ks¯) → H q (Xη¯, Kη¯) → H q (Xs¯, RΦ(K)) → · · · . So H · (Xs¯, RΦ(K)) measures the difference between H · (Xs¯, Ks¯) and H · (Xη¯, Kη¯) if X is proper over S. Let f : X → X 0 be an S-morphism of S-schemes. Fix notation by the following commutative diagram: ˜

˜i j e = X ×S S˜ ← Xη¯ → X Xs¯ fη¯ ↓ ↓ f˜ ↓ fs¯ ˜ ˜i0 j0 e 0 = X 0 ×S S˜ ← Xs¯0 . Xη¯0 → X

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For any K ∈ ob D+ (X, A), we define a morphism RΨη ((Rf∗ K)η ) → Rfs¯∗ RΨη (Kη ) to be the composite of the canonical morphisms e RΨη ((Rf∗ K)η ) ∼ = ˜i0∗ R˜j∗0 ˜j 0∗ Rf˜∗ K

e → ˜i0∗ R˜j∗0 Rfη¯∗ ˜j ∗ K ∼ e = ˜i0∗ Rf˜∗ R˜j∗ ˜j ∗ K

e → Rfs¯∗˜i∗ R˜j∗ ˜j ∗ K

= Rfs¯∗ RΨη (Kη ).

+ ob Dtor (X),

When f is proper and K ∈ this is an isomorphism by the proper base change theorem 7.3.1. Suppose that f : X → X 0 is an open immersion. Define a morphism fs¯! RΨη (Kη ) → RΨη ((f! K)η ) to be the composite of the canonical morphisms e fs¯! RΨη (Kη ) ∼ = fs¯!˜i∗ R˜j∗ ˜j ∗ K

∼ e = ˜i0∗ f˜! R˜j∗ ˜j ∗ K e → ˜i0∗ R˜j∗0 fη¯! ˜j ∗ K

∼ e = ˜i0∗ R˜j∗0 ˜j 0∗ f˜! K ∼ = RΨη ((f! K)η ).

+ Suppose that K ∈ ob Dtor (X), f : X → X 0 is an S-compactifiable morphism, and k

f¯

X ,→ X → X 0

is a compactification for f , where k is an open immersion, and f¯ is a proper S-compactifiable morphism. Define a morphism Rfs¯! RΨη (Kη ) → RΨη ((Rf! K)η ) to be the composite of Rfs¯! RΨη (Kη ) ∼ = Rf¯s¯∗ ks¯! RΨη (Kη ) → Rf¯s¯∗ RΨη ((k! K)η ) ∼ = RΨη ((Rf¯∗ k! K)η ) ∼ = RΨη ((Rf! K)η ).

If f is proper, this is the inverse of the morphism RΨη ((Rf∗ K)η ) → Rfs¯∗ RΨη (Kη )

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defined above. Suppose that X → S is compactifiable. Applying the above construction to the structure morphism X → S, we get a homomorphism Hcq (Xs¯, RΨη (Kη )) → Hcq (Xη¯ , Kη¯) for each q. If X is proper over S, it is the inverse of the canonical isomorphism H q (Xη¯ , Kη¯) → H q (Xs¯, RΨη (Kη )). For any σ ∈ I in the inertia group, we have commutative diagrams H q (Xη¯, Kη¯) ↓

σ−id

Hcq (Xη¯, Kη¯) ↑

σ−id

var(σ)

H i (Xη¯ , Kη¯) ↓

var(σ)

Hci (Xη¯ , Kη¯) ↑

→

H q (Xs¯, RΨη (Kη )) → H q (Xs¯, RΦ(K)) → H q (Xs¯, RΨη (Kη )), →

Hcq (Xs¯, RΨη (Kη )) → Hcq (Xs¯, RΦ(K)) → Hcq (Xs¯, RΨη (Kη )). For any L ∈ D+ (X 0 ), define a morphism fs¯∗ RΨη (Lη ) → RΨη ((f ∗ L)η ) to be the composite of the canonical morphisms e fs¯∗ RΨη (Lη ) = fs¯∗˜i0∗ R˜j∗0 ˜j 0∗ L ∼ e = ˜i∗ f˜∗ R˜j∗0 ˜j 0∗ L e → ˜i∗ R˜j∗ fη¯∗ ˜j 0∗ L

∼ e = ˜i∗ R˜j∗ ˜j ∗ f˜∗ L ∗ ∼ = RΨη ((f L)η ).

If f is smooth and L ∈ D+ (X 0 , Z/n) for some n invertible on S, this is an isomorphism by the smooth base change theorem 7.7.2. Proposition 9.2.3. Suppose that X is smooth over S, and H q (K) are locally constant sheaves of Z/n-modules for some n invertible on S. Then RΦ(K) = 0. Proof. We need to show that Ks¯ → RΨη (Kη ) is an isomorphism. The problem is local with respect to the etale topology on X. So we may assume

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that K is a constant sheaf associated to a Z/n-module M . Fix notation by the following diagram ˜

˜i j e = X ×S S˜ ← Xη¯ → X Xs¯ fη¯ ↓ f˜ ↓ ↓ fs¯ j i η¯ → S˜ ← s¯. Denote by M the constant sheaf on S. We have a commutative diagram fs¯∗ i∗ M → fs¯∗ RΨη (M ) ∼ ↓ =↓ ˜i∗ f˜∗ M → RΨη (f˜∗ M ).

The top horizontal arrow is an isomorphism by 9.2.2 (iii). The right vertical arrow is an isomorphism by the smooth base change theorem 7.7.2. So ˜i∗ f˜∗ M → RΨη (f˜∗ M )

is an isomorphism.

9.3

Generic Base Change Theorem and Generic Local Acyclicity

([SGA 4 21 ] Th. finitude 2.) Throughout this section, A is a noetherian ring such that nA = 0 for some integer n invertible on a base scheme S. The main results of this section are the following theorems. Theorem 9.3.1. Let S be a noetherian scheme, X and Y two S-schemes of finite type, f : X → Y an S-morphism, and F a constructible sheaf of A-modules on X. Then there exists a dense open subset U of S such that the following conditions hold: (i) (Rq f∗ F )|YU are constructible and are nonzero only for finitely many q, where YU is the inverse image of U in Y . (ii) The formation of Rq f∗ F commutes with any base change S 0 → U ⊂ S, or equivalently, the formation of Rq fU∗ (F |XU ) commutes with any base change S 0 → S, where XU is the inverse image of U in X, and fU : XU → YU is the morphism induced by f . Theorem 9.3.2. Let S be a noetherian scheme, f : X → S a morphism of b finite type, and K ∈ ob Dctf (X, A). Then there exists a dense open subset U of S such that f |f −1 (U) : f −1 (U ) → U is universally locally acyclic relative to K|f −1 (U) .

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When S is the spectrum of a field, we must have U = S. So we have the following: Corollary 9.3.3. Let X and Y be schemes of finite type over a field k, let f : X → Y be a k-morphism, and let F be a constructible sheaf of Amodules. Then Rq f∗ F are constructible for all q. If k is separably closed, then H q (X, F ) are finitely generated A-modules. Corollary 9.3.4. Let k be a field, and let f : X → Spec k be a morphism of finite type. Then f is universally strongly locally acyclic relative to any b K ∈ ob Dctf (X, A). Corollary 9.3.5 (K¨ unneth formula). Let k be a field, let Xi and Yi (i = 1, 2) be k-schemes of finite type, let fi : Xi → Yi be k-morphisms, and let pi : X1 ×k X2 → Xi and qi : Y1 ×k Y2 → Yi be projections. For any b Ki ∈ ob Dctf (Xi , A), we have q ∗ Rf1∗ K1 ⊗L q ∗ Rf2∗ K2 ∼ = R(f1 × f2 )∗ (p∗ K1 ⊗L p∗ K2 ). 1

A

2

1

A

2

Proof. Fix notation by the following commutative diagram of Cartesian squares: f 00

b00

f0

q2

1 1 X1 × k X2 → Y1 ×k X2 → f200 ↓ f20 ↓

1 X1 ×k Y2 → Y1 ×k Y2 → b00 ↓ q1 ↓ 2

X1

f1

→

b

X2 ↓ f2 Y2 ↓ b2

1 → Spec k.

Y1

b By 6.5.6, 7.5.6 and 9.3.3, we have Rf1∗ K1 ∈ ob Dctf (Y1 , A). By 7.5.7 and 7.6.2 and 9.3.4, b1 is universally strongly locally acyclic relative to Rf1∗ K. By 7.6.9, we have an isomorphism q ∗ Rf1∗ K1 ⊗L q ∗ Rf2∗ K2 ∼ = Rf 0 (f 0∗ q ∗ Rf1∗ K1 ⊗L b00∗ K2 ). 1

A

2

2∗

2

1

A 1

Similarly, b2 f2 is universally strongly locally acyclic relative to K2 and we have an isomorphism f 0∗ q ∗ Rf1∗ K1 ⊗L b00∗ K2 ∼ = Rf 00 (f 00∗ b00∗ K1 ⊗L f 00∗ b00∗ K2 ), 2

1

A 1

1∗

2

2

A

1

1

and hence 0 00∗ 0 00 00∗ 00∗ L 00∗ 00∗ ∼ Rf2∗ (f20∗ q1∗ Rf1∗ K1 ⊗L A b1 K2 ) = Rf2∗ Rf1∗ (f2 b2 K1 ⊗A f1 b1 K2 ).

It follows that ∗ 0 00 00∗ 00∗ L 00∗ 00∗ ∼ q1∗ Rf1∗ K1 ⊗L A q2 Rf2∗ K2 = Rf2∗ Rf1∗ (f2 b2 K1 ⊗A f1 b1 K2 ).

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Proof of 9.3.1. Working with each irreducible component of S, we may assume that S is an integral scheme. Let η be its generic point. First consider the case where X is smooth over S pure of relative dimension d, A = Z/m for some integer m invertible on S, F and Rq f! F ∨ are locally constant for all q, and Y = S, where F ∨ = H om(F , Z/m). By the Poincar´e duality (8.4.6 and 8.5.2), we have Rf∗ RH om(F ∨ , Z/m(d)[2d]) ∼ = RH om(Rf! F ∨ , Z/m). Since F and Rq f! F ∨ are locally constant, and Z/m is an injective Z/mmodule, we have RH om(F ∨ , Z/m(d)[2d]) ∼ = H om· (F ∨ , Z/m(d)[2d]) ∼ = F (d)[2d], −q ∨ q ∨ ∼ R H om(Rf! F , Z/m) = H om(R f! F , Z/m). It follows that R2d−q f∗ F ∼ = H om(Rq f! F ∨ , Z/m)(−d). For each q, the right-hand side is locally constant and constructible (7.8.1), commutes with any base change (7.4.4), and is nonzero only for finitely many q (7.4.5). So 9.3.1 holds for U = S. Next we consider the case where X is smooth over S, Y = S, F is locally constant, and there exists a galois etale covering space X1 → X such that the inverse image F1 of F in X1 is a constant sheaf, say associated to an A-module F . Working with each connected component of X, we may assume that X is pure of relative dimension d over S. We have M F = F` , `

where ` goes over the set of prime numbers invertible on S, and F` is the `-torsion part of F . Working with each direct factor F` , we may assume that `k F = 0 for some integer k and some prime number ` invertible on S. Filtrating F by li F (i ∈ N ∪ {0}) and working with each successive quotient `i F /`i+1 F , we may assume `F = 0. Replacing A by A/`A, we may assume `A = 0. Let f1 : X1 → S be the composite of X1 → X and f . By 6.5.5, we have ∗ L ∼ q ∼ q Rq f1∗ F1 ∼ = Rq f1∗ (Z/` ⊗L Z/` f1 F ) = R f1∗ Z/` ⊗Z/` F = R f1∗ Z/` ⊗Z/` F.

Since Rq f1! Z/` are constructible, and are nonzero only for finitely many q, by shrinking S, we may assume they are locally constant. By our previous discussion, Rq f1∗ Z/` are locally constant, constructible, nonzero for only finitely many q, and commute with any base change. Rq f1∗ F1 have

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the same property. Let G = Aut(X1 /X). We have the Hochschild–Serre spectral sequence E2pq = H p (G, Rq f1∗ F1 ) ⇒ Rp+q f∗ F . So Rq f∗ F are locally constant, constructible, and commute with any base change. In particular, we have (Rq f∗ F )η¯ ∼ = H q (Xη¯, F ). Since H q (Xη¯, F ) are nonzero only for finitely many q (7.5.5), Rq f∗ F have the same property. To prove 9.3.1 in the general case, the problem is local on Y and we may assume that Y is affine. Covering X by finitely many affine open subsets and applying 5.6.10, we are reduced to the case where X is also affine. Then we can write f = f¯j with j being an open immersion and f¯ being proper. 9.3.1 holds for the proper morphism f¯. So it suffices to prove 9.3.1 for any open immersion f . As 9.3.1 holds for any closed immersion, it suffices to consider the case where f is an open immersion with dense image. We prove this by induction on dim Xη . Suppose that dim Xη ≤ 0 and f is an open immersion with dense image. We then have Xη = Yη . Indeed, let Y1 , . . . , Ym be the irreducible components of Y , and let η1 , . . . , ηm be the corresponding generic points. Since X is dense in Y , η1 , . . . , ηm all lie in X. If Yiη 6= Ø, then ηi must lie in Yiη . Note that ηi is the generic point of Yiη . As ηi ∈ X, the morphism (X ∩ Yi )η → Yiη is an open immersion with dense image. But dim (X ∩ Yi )η = 0. So we must have (X ∩ Yi )iη = Yiη . It follows that Xη = Yη . Shrinking S, we may assume X = Y . 9.3.1 holds trivially in this case. Suppose that 9.3.1 holds for any immersion so that the fiber over η of its domain has dimension ≤ d − 1. Let us prove that it holds for any immersion f : X → Y with dim Xη ≤ d. We first show that after shrinking S, there exists an open subset V ⊂ Y such that Y − V → S is finite, and that Rq fV ∗ F are constructible, nonzero only for finitely many q, and commute with any base change S 0 → S, where fV : f −1 (V ) → V is the morphism induced by f . Cover Y by finitely many affine open subsets Yi . Choose immersions Yi ,→ AkS . Let πj : AkS → A1S (j = 1, . . . , k) be the projections. By 7.5.3, the generic fiber of (πj |Yi ) ◦ (fYi ) has dimension ≤ d − 1 for each pair (i, j), and we can apply the induction hypothesis to the morphism fYi : f −1 (Yi ) → Yi over the base A1S , where Yi is considered as an A1S -scheme through the morphism πj |Yi . We can find dense open

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subsets Uij of A1S so that (Rq f∗ F )|Yi ∩π−1 (Uij ) are constructible, nonzero j

only for finitely many q, and commute with any base change S 0 → A1S . Let S V = i,j (Yi ∩ πj−1 (Uij )). Then Rq fV ∗ F are constructible, nonzero only for finitely many q, and commute with any base change S 0 → S. We have [ Y −V ⊂ Yi ∩ (AkS − ∪j πj−1 (Uij )) i

=

[ i

(A1S

Yi ∩ ((A1S − Ui1 ) ×S · · · ×S (A1S − Uik )) .

As the fiber of − Ui1 ) ×S · · · ×S (A1S − Uik ) → S over η is finite, we may assume that Y − V → S is finite by shrinking S. Let us prove that 9.3.1 holds for the immersion f : X → Y under the extra condition that X is smooth over S, F is locally constant, and there exists a galois etale covering space X1 → X such that the inverse image of F in X1 is a constant sheaf. The problem is local with respect to Y . We may assume that Y is affine. Then Y is an open subscheme of a scheme projective over S. We are thus reduced to the case where Y is projective over S. Shrinking S, we may find an open subset V of Y such that Y − V → S is finite, and Rq fV ∗ F are constructible, nonzero only for finitely many q, and commute with any base change S 0 → S. Fix notation by the following commutative diagram:

X

f

V j ↓

i

→ Y ← Y − V, a& b↓ .c S

where i and j are immersions, and a, b, c are the structure morphisms. By the discussion at the beginning of the proof, under our extra condition, after shrinking S, Rq a∗ F are constructible, nonzero only for finitely many q, and commute with any base change. By the choice of V , the proper base change theorem 7.3.1, 7.4.5, and 7.8.1, after shrinking S, Rq b∗ j! j ∗ Rf∗ F are constructible, nonzero only for finitely many q, and commute with any base change. Applying Rb∗ to the distinguished triangle j! j ∗ Rf∗ F → Rf∗ F → i∗ i∗ Rf∗ F →, we get a distinguished triangle Rb∗ j! j ∗ Rf∗ F → Ra∗ F → c∗ i∗ Rf∗ F → .

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It follows that after shrinking S, c∗ i∗ Rq f∗ F are constructible, nonzero only for finitely many q, and commute with any base change. As c is finite, i∗ Rq f∗ F have the same properties. By our choice of V , j! j ∗ Rq f∗ F also have these properties. So Rq f∗ F have these properties. This proves that the theorem holds for f under our extra condition. Finally we prove that 9.3.1 holds for the immersion f : X → Y unconditionally. We may assume that there exists an open dense subset W of X smooth over S. Indeed, if S is the spectrum of a perfect field, it suffices to replace X by Xred . In general, we use the passage to limit argument, and we need to shrink S, make a finite radiciel surjective base change, and replace X by a closed subscheme defined by a nilpotent ideal. These changes have no effect on etale cohomology. Shrinking W , we may assume that F |W is locally constant, and there exists a galois etale covering space W1 → W such that the inverse image of F in W1 is a constant sheaf (5.8.1 (ii)). Let j : W ,→ X be the open immersion. Choose a distinguished triangle F → Rj∗ j ∗ F → ∆ → . The cohomology sheaves of ∆ are supported in X −W . We have shown that the theorem holds for the open immersion j : W → X. So after shrinking S, we may assume that Rq j∗ j ∗ F are constructible, nonzero only for finitely many q, and commutes with any base change. It follows that H q (∆) have the same property. Since dim (X −W )η ≤ d−1, by the induction hypothesis applied to X − W → Y and ∆|X−W , we see that Rq f∗ ∆ are constructible, nonzero only for finitely many q, and commute with any base change. We have a distinguished triangle Rf∗ F → R(f j)∗ j ∗ F → Rf∗ ∆ → . We have shown that the theorem holds for the immersion f j : W → Y . So after shrinking S, Rq (f j)∗ j ∗ F are constructible, nonzero only for finitely many q, and commute with any base change. It follows that Rq f∗ F have the same property. So 9.3.1 holds for f . Lemma 9.3.6. Let f : X → S be a proper morphism of noetherian b schemes, let K be an object in Dctf (X, A) such that Rq f∗ K are locally constant for all q, and let W be an open subset of X such that f |W : W → S is locally acyclic relative to K and such that X − W is finite over S. Then f is locally acyclic relative to K. Proof. Let s ∈ S, let Ses¯ be the strict localization of S at s¯, and let j : t → Ses¯ be an algebraic geometric point. Fix notation by the following

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diagram ˜ j

˜i

Xt = X ×S t → X ×S Ses¯ ← Xs¯ = X ×S s¯ ft ↓ f˜ ↓ ↓ fs¯ j i t→ Ses¯ ← s¯,

where vertical arrows are base changes of f . Denote the inverse image e We need to show that ˜i∗ K e → ˜i∗ R˜j∗ ˜j ∗ K e is an of K on X ×S Ses¯ by K. isomorphism. Let e → ˜i∗ R˜j∗˜j ∗ K e →∆→ ˜i∗ K

be a distinguished triangle. We need to show that ∆ is acyclic. Since f is locally acyclic relative to K on W , the cohomology sheaves of ∆ are supported on (X − W )s¯, which is a finite set. To prove that ∆ is acyclic, it suffices to show that RΓ(Xs¯, ∆) is acyclic. We have a distinguished triangle e → RΓ(Xs¯, ˜i∗ R˜j∗ ˜j ∗ K) e → RΓ(Xs¯, ∆) → . RΓ(Xs¯, ˜i∗ K)

It suffices to show that the canonical morphism

e → RΓ(Xs¯, ˜i∗ R˜j∗ ˜j ∗ K) e RΓ(Xs¯, ˜i∗ K)

is an isomorphism. For each q, since Rq f∗ K is locally constant, the specialization homomorphism (Rq f∗ K)s¯ → (Rq f∗ K)t¯ is an isomorphism. By 5.9.5, we have e (Rq f∗ K)s¯ ∼ = H q (X ×S Ses¯, K),

and by the proper base change theorem 7.3.1, we have e (Rq f∗ K)t¯ ∼ = H q (Xt¯, ˜j ∗ K).

Through these isomorphisms, the specialization homomorphism is identified with the canonical homomorphism e → H q (Xt¯, ˜j ∗ K). e H q (X ×S Ses¯, K)

Since the specialization homomorphism is an isomorphism, we have e ∼ e H q (X ×S Ses¯, K) = H q (Xt¯, ˜j ∗ K).

We have a commutative diagram of canonical morphisms e e → H q (Xs¯, ˜i∗ R˜j∗ ˜j ∗ K) e ← H q (X ×S Ses¯, R˜j∗ ˜j ∗ K) H q (Xt¯, ˜j ∗ K) k ↑ ↑ e ← e e H q (Xt¯, ˜j ∗ K) H q (X ×S Ses¯, K) → H q (Xs¯, ˜i∗ K).

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We have just shown that the lower horizontal arrow in the square on the left of the diagram is an isomorphism. By the proper base change theorem 7.3.3, the two horizontal arrows in the square on the right of the diagram are isomorphisms. The canonical morphism e → H q (Xt¯, ˜j ∗ K) e H q (X ×S Ses¯, R˜j∗ ˜j ∗ K)

is also an isomorphism. It follows that all arrows in the above diagram are isomorphisms. So e → H q (Xs¯, ˜i∗ R˜j∗˜j ∗ K) e H q (Xs¯, ˜i∗ K)

is an isomorphism. Hence

is an isomorphism.

e → RΓ(Xs¯, ˜i∗ R˜j∗ ˜j ∗ K) e RΓ(Xs¯, ˜i∗ K)

Proof of Theorem 9.3.2. We may assume that S is an integral scheme with generic point η, and we use induction on dim Xη . First we use the induction hypothesis to prove the following statement: (*) After shrinking S, we can find an open subset W of X such that f |W : W → S is universally locally acyclic relative to K|W and X − W → S is finite. Cover X by finitely many affine open subsets Xi (i = 1, . . . , m) so that there exist immersions Xi ,→ AkS . Let πj : AkS → A1S (j = 1, . . . , k) be the projections. For each pair (i, j), the generic fiber of πj |Xi has dimension ≤ dim Xη − 1 by 7.5.3. By the induction hypothesis, we can find a dense open subset Vij of A1S such that πj : Xi ∩ πj−1 (Vij ) → Vij is universally locally acyclic relative to K. The projection Vij → S is smooth and hence universally strongly locally acyclic relative to A by 7.7.5. So by 7.7.6, f : Xi ∩ πj−1 (Vij ) → S is universally locally acyclic relative to K. Let S W = i,j (Xi ∩ πj−1 (Vij )). We have [ X −W ⊂ Xi ∩ AkS − ∪j πj−1 (Vij ) i

=

[ i

Xi ∩ (A1S − Vi1 ) ×S · · · ×S (A1S − Vik ) .

As the generic fiber of (A1S − Vi1 ) ×S · · · ×S (A1S − Vik ) → S is finite, we may assume that X − W → S is finite by shrinking S. This proves (*) under the induction hypothesis. Now let us prove the theorem. The problem is local. We may assume that S is affine. Covering X by finitely many affine open subsets, and

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working with each of them, we may assume that X is affine. Then X is an open subscheme of a proper S-scheme. So it suffices prove the theorem for any proper S-scheme X. By 7.4.5 and 7.8.1, after shrinking S, we may assume that Rq f∗ K are locally constant for all q. By (*), we may assume that there exists an open subset W of X such that f |W : W → S is universally locally acyclic relative to K, and X − W → S is finite. We then apply 9.3.6.

9.4

Finiteness of RΨη

([SGA 4 21 ] Th. finitude 3.) The main result of this section is the following: Theorem 9.4.1. Let S be a strictly local trait, s its closed point, η its generic point, and A a noetherian ring such that nA = 0 for some integer n invertible on S. For any S-scheme X of finite type and any constructible sheaf of A-modules F on Xη , Rq Ψη F are constructible for all q. Before proving the theorem, we make some preparations. Let S = Spec R, where R is a strictly henselian discrete valuation ring with maximal ideal m. Let s0 be the generic point of A1s = Spec (R/m)[t], and denote the image of s0 in A1S = Spec R[t] also by s0 . It corresponds to the prime ideal m[t] of R[t]. Let S 0 be the strict localization of A1S at s¯0 . Then S 0 = Spec R0 for a strictly henselian discrete valuation ring R0 , and any uniformizer of R is also a uniformizer of R0 . Let η 0 be the generic point of S 0 . We have Spec k(η 0 ) ∼ = Spec k(η) ×S S 0 . 0 0 Let K = k(¯ η ) ⊗k(η) k(η ). Then K 0 is a field. To show this, it suffices to show that for any finite galois extension K1 of k(η) contained in k(¯ η ), K1 ⊗k(η) k(η 0 ) is a field. Let R1 be the integral closure of R in K1 and let S1 = Spec R1 . Then R1 is a strictly henselian discrete valuation ring with fraction field K1 . Since the residue field of R1 is purely inseparable over the residue field of R, there exists one and only one point in S1 ×S S 0 lying above the closed point of S 0 . On the other hand, S1 ×S S 0 is finite over S 0 . It follows that S1 ×S S 0 is strictly local, and it is the strict localization of A1S1 at the generic point of the special fiber of A1S1 → S1 . We have Spec (K1 ⊗k(η) k(η 0 )) ∼ = Spec K1 ×Spec k(η) (Spec k(η) ×S S 0 ) ∼ = Spec K1 ×S (S1 ×S S 0 ). 1

So K1 ⊗k(η) k(η 0 ) is the function field of S1 ×S S 0 . We have Gal(K1 ⊗k(η) k(η 0 )/k(η 0 )) ∼ = Gal(K1 /k(η)).

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It follows that K 0 = k(¯ η ) ⊗k(η) k(η 0 ) is a field and Gal(K 0 /k(η 0 )) ∼ η )/k(η)). = Gal(k(¯ 0

0

Let K be a separable closure of K 0 and let η¯0 = Spec K . Note that 0 Gal(K /K 0 ) is a pro-p-group, where p is the characteristic of the residue field of R. To see this, we need to show that any finite tamely ramified extension L of k(η 0 ) is contained in K 0 . Let π be a uniformizer of R. √ It is also a uniformizer of R0 . By 8.1.3, we have L = k(η 0 )[ n π], where √ η ), we have L ⊂ K 0 . n = [L : k(η 0 )]. Since n π ∈ k(¯ Lemma 9.4.2. Notation as above. Let X 0 be an S 0 -scheme, let F be a sheaf of Z/n-modules on Xη0 0 ∼ = Xη0 for some integer n invertible on S, and let RΨη0 (F ) and RΨη (F ) be the nearby cycles of F with respect to X 0 /S 0 and X 0 /S, respectively. We have Rq Ψη (F ) ∼ = Rq Ψη0 (F )Gal(K Proof.

0

/K 0 )

.

Fix notation by the following commutative diagram j1 j2 j3 i Xη¯0 0 → Xη¯0 → Xη0 ∼ = Xη0 0 → X 0 ← Xs0 ∼ = Xs0 0 ↓ ↓ ↓ ↓ ↓ 0 0 0 0 η¯ → Spec K → η → S ← S 0 ×S s ∼ = s0 ↓ ↓ ↓ ↓ η¯ → η → S ← s.

We have Ru Ψη (F ) = i∗ Ru (j3 j2 )∗ j2∗ F ,

Ru Ψη0 (F ) = i∗ Ru (j3 j2 j1 )∗ j1∗ j2∗ F .

We have the Hochschild–Serre spectral sequence 0

E2uv = H u (Gal(K /K 0 ), Rv Ψη0 (F )) ⇒ Ru+v Ψη (F ). 0

Since Gal(K /K 0 ) is a pro-p-group and Rv Ψη0 (F ) are n-torsion sheaves and n is relatively prime to p, we have H u (Gal(K 0 /K 0 ), Rv Ψη0 (F )) = 0 0

0

for all u ≥ 1. (The functor G → G Gal(K /K ) is exact in the category of 0 torsion sheaves with continuous Gal(K /K 0 )-action and with torsion prime to p.) So the spectral sequence degenerates, and we have Rq Ψη (F ) ∼ = Rq Ψη0 (F )Gal(K

0

/K 0 )

.

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1 Lemma 9.4.3. Let k be a field, πi : Am k → Ak (i = 1, . . . , m) the projections, η¯ an algebraic geometric point over the generic point of A1k , X a subscheme of Am η k , and F a sheaf of A-modules on X. For each i, let Xi¯ be the fiber of πi |X over η¯, and let Fi¯η be the inverse image of F on Xi¯η . Suppose Fi¯η are constructible. Then there exists a constructible subsheaf F 0 of F such that every section of the sheaf F /F 0 has finite support.

Proof. By 5.9.8, for each i, we can find an etale neighborhood U of η¯ in A1k , a constructible sheaf GiU on XiU = X ×πi |X ,A1k U such that the inverse image Gi¯η of GiU on Xi¯η is isomorphic to Fi¯η . Moreover, we may assume that the isomorphism is induced by a morphism GiU → FiU , where FiU is the inverse image of F on XiU . Let φ : XiU → X be the projection. Then the above morphism induces a morphism φ! GiU → F . Let Fi0 be the image of this morphism. It is constructible. The image of the morphism φ∗ φ! GiU → FiU is φ∗ Fi0 . Since the morphism GiU → FiU can be factorized as the composite GiU → φ∗ φ! GiU → FiU , φ∗ Fi0 contains the image of GiU → FiU . As Gi¯η → Fi¯η is an isomorphism, the inverse image of F /Fi0 on Xiη is 0. Let F 0 be the image of the canonical morphism n M i=1

Fi0 → F .

Then F 0 is constructible, and the inverse image of F /F 0 on Xiη is 0 for each i. Any section of the sheaf F /F 0 has finite support. Indeed, let f : V → X be an etale X-scheme of finite type and let s ∈ (F /F 0 )(V ). By 5.9.3, for each i, we can find a nonempty open subset Ui of A1k such that the restriction of s to (πi |X ◦ f )−1 (Ui ) is 0. We have V − As

m [

i=1

(A1k

(πi |X ◦ f )−1 (Ui ) = f −1 X ∩ (A1k − U1 ) ×k · · · ×k (A1k − Un ) .

− U1 ) ×k · · · ×k (A1k − Un ) is finite, the support of s is finite.

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Proof of 9.4.1. We use induction on dim Xη . When dim Xη < 0, we have Xη = Ø. Then Rq Ψη (F ) = 0 for all q. Suppose that the theorem holds for those S-schemes whose generic fibers have dimensions < d. Let X be an S-scheme with dim Xη = d. We first prove that for each q, there exists a constructible subsheaf Gq of Rq Ψη (F ) such that the support of any section of Rq Ψη (F )/G is finite. Cover X by finitely many affine open subsets Xi , and let ji : Xis → Xs be the open immersion. If we can find constructible subsheaves Gqi of Rq Ψη (F )|Xis such that the sections of Rq Ψη (F )|Xis /Gqi have finite support, then we can take Gq to be the image of the canonical morphism M ji! Gqi → Ri Ψη (F ). i

So we may assume that X is affine. Then we have an immersion X ,→ Am S. 1 0 Let π be one of the projections πi |X : X ,→ Am → A , let S be the strict S S localization of A1S at the generic point of the special fiber of A1S → S, let X 0 = X ×π,A1S S 0 , let λ : X 0 → X be the projection, and let F 0 be the inverse image of F on X 0 . X 0 → S0 λ ↓ ↓ & π X → A1S → S.

Since X 0 is the inverse limit of an inverse system of etale X-schemes, we have λ∗s Rq Ψη (F ) = Rq Ψη (F 0 ). By 9.4.2, we have Rq Ψη (F 0 ) = Rq Ψη0 (F 0 )Gal(K

0

/K 0 )

.

By the induction hypothesis, Rq Ψη0 (F 0 ) are constructible. So λ∗s Rq Ψη (F ) are constructible. By 9.4.3, there exists a constructible subsheaf Gq of Rq Ψη (F ) such that the support of any section of Rq Ψη (F )/Gq is finite. Let us prove that Rq Ψη (F ) are constructible. The problem is local on X. We may assume that X is affine. Then X is an open subscheme of a proper S-scheme. So it suffices to prove the theorem for any proper S-scheme X. We then have a biregular spectral sequence E2pq = H p (Xs , Rq Ψη (F )) ⇒ H p+q (Xη¯, Fη¯ ). Let Hq = Rq Ψη (F )/Gq . Since sections of Hq have finite support, we have H p (Xs , Hq ) = 0 for all p ≥ 1 by 7.2.4. To prove that Rq Ψη (F ) are

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constructible, it suffices to show that H 0 (Xs , Hq ) are finitely generated Amodules. For any two A-modules M and N , we write M ∼ N if there exists a finite family of A-modules M0 , . . . , Mk such that M0 = M , Mk = N , and for each i ∈ {0, . . . , k − 1}, there exists a homomorphism Mi → Mi+1 or a homomorphism Mi+1 → Mi with finite kernel and finite cokernel. Since Gq are constructible, H p (Xs , Gq ) are finitely generated A-modules by 7.8.1. From the long exact sequences of cohomology groups associated to the short exact sequence 0 → Gq → Rq Ψη (F ) → Hq → 0, we get E2pq ∼ H p (X, Hq ). In particular, E2pq ∼ 0 for all p 6= 0, that is, the spectral sequence degenerates modulo the equivalence relation ∼. So we have E20q ∼ H q (Xη¯ , Fη¯ ). By 7.8.1, we have H q (Xη¯, Fη¯ ) ∼ 0. So we have H 0 (Xs , Hq ) = E20q ∼ 0.

This proves our assertion.

9.5

Finiteness Theorems

([SGA 4 21 ] Th. finitude 1, 3.) Throughout this section, A is a noetherian ring such that nA = 0 for some integer n invertible on a base scheme S. Theorem 9.5.1. Suppose that S is a regular noetherian scheme of dimension ≤ 1. Let X and Y be S-schemes of finite type, f : X → Y an Smorphism, and F a constructible sheaf of A-modules on X. Then Rq f∗ F are constructible for all q, and are nonzero only for finitely many q. Proof. First suppose that S is a strictly local trait. Let η and s be the generic point and the closed point of S, respectively. We first prove that for any S-scheme X of finite type and any constructible sheaf of A-modules Fη on Xη , the sheaves i∗ Rq j∗ Fη are constructible for all q and are nonzero only for finitely many q, where i : Xs → X and j : Xη → X are immersions. We have the Hochschild–Serre spectral sequence E2uv = H u (Gal(¯ η /η), Rv Ψη (Fη )) ⇒ i∗ Ru+v j∗ F .

It suffices to show that H u (Gal(¯ η /η), Rv Ψη (Fη )) are constructible for all u, v and are nonzero only for finitely many pairs (u, v). Let P be the wild inertia subgroup of Gal(¯ η /η). It is a pro-p-subgroup of Gal(¯ η /η) and ∼ Gal(¯ η /η)/P = lim µn (k(η)), ←− (n,p)=1

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where p is the characteristic of the residue field of S. Let σ ∈ Gal(¯ η /η)/P be a topological generator so that its components in each µn (k(η)) is a primitive n-th root of unity. Since Rq Ψη (Fη ) are n-torsion sheaves and n is relatively prime to p, we have H v (P, Rq Ψη (Fη )) = 0 for any v 6= 0. (The functor G → G P is exact in the category of torsion sheaves with continuous P -action and with torsion prime to p.) So the Hochschild–Serre spectral sequence η /η)/P, H v (P, Rq Ψη (Fη ))) ⇒ H E2uv = H u (Gal(¯

u+v

(Gal(¯ η /η), Rq Ψη (Fη ))

degenerates, and we have

H u (Gal(¯ η /η), Rq Ψη (Fη )) ∼ η /η)/P, (Rq Ψη (Fη ))P ). = H u (Gal(¯ By 4.3.9, we have

 ker(σ − 1, (Rq Ψη (Fη ))P ) if u = 0, u q P ∼ H (Gal(¯ η /η)/P, (R Ψη (Fη )) ) = coker(σ − 1, (Rq Ψη (Fη ))P ) if u = 1,  0 if u 6= 0, 1. By 9.4.1 and 9.2.2 (ii), Rq Ψη (Fη ) are constructible for all q, and are nonzero only for finite many q. Hence ker(σ − 1, (Rq Ψη (Fη ))P ) and coker(σ − 1, (Rq Ψη (Fη ))P ) are constructible for all q, and are nonzero only for finitely many q. This proves our assertion. To prove the theorem, we may assume that S is connected and hence integral. If dim S = 0, the theorem follows from 9.3.3. Suppose dim S = 1. Then by 9.3.1, there exists a finite closed subset T of S such that over S −T , Rq f∗ F are constructible for all q and are nonzero only for finitely many q. To prove the theorem, it suffices to show that Rq f∗ F |Ys are constructible for all q and are nonzero only for finitely many q for all s ∈ T . The problem is local on S. Replacing S by the strict localization of S at s, we may assume that S is a strictly local trait. Fix notation by the following commutative diagram. j1

i

j2

i

1 Xη → X ← Xs fη ↓ f ↓ ↓ fs 2 Yη → Y ← Ys ↓ ↓ ↓ η → S ← s.

We need to show that i∗2 Rq f∗ F are constructible for all q and are nonzero only for finitely many q. Choose a distinguished triangle F → Rj1∗ j1∗ F → ∆ → .

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We then have a distinguished triangle i∗2 Rf∗ F → i∗2 Rf∗ Rj1∗ j1∗ F → i∗2 Rf∗ ∆ → . To prove our assertion, it suffices to prove that i∗2 Rq f∗ ∆ and i∗2 Rq f∗ Rj1∗ j1∗ F are constructible for all q and are nonzero only for finitely many q. We have shown that i∗1 Rq j1∗ j1∗ F have these properties. So the cohomology sheaves of i∗1 ∆ are all constructible and only finite many of them are nonzero. By 9.3.3, Rq fs∗ i∗1 ∆ are constructible for all q and are nonzero only for finite many q. The cohomology sheaves of ∆ are supported on Xs . So we have i∗2 Rf∗ ∆ ∼ = i∗2 Rf∗ i1∗ i∗1 ∆ ∼ = i∗2 i2∗ Rfs∗ i∗1 ∆ ∼ = Rfs∗ i∗1 ∆. Hence i∗2 Rq f∗ ∆ are constructible for all q and are nonzero only for finitely many q. We have i∗2 Rf∗ Rj1∗ j1∗ F ∼ = i∗2 Rj2∗ Rfη∗ j1∗ F . By 9.3.3, the cohomology sheaves of Rfη∗ j1∗ F are all constructible and only finitely many of them are nonzero. By our previous discussion, i∗2 Rq j2∗ Rfη∗ j1∗ F are constructible for all q and are nonzero only for finitely many q. This proves our assertion. Theorem 9.5.2. Suppose that S is a regular noetherian scheme of dimension ≤ 1. Let X and Y be S-schemes of finite type and let f : X → Y be an S-morphism. Then Rf∗ , Rf! , f ∗ , Rf ! have finite cohomological dimensions, b b Rf∗ and Rf! map objects in Dc (X, A) (resp. Dtf (X, A), resp. Dctf (X, A)) b b ∗ to objects in Dc (Y, A) (resp. Dtf (Y, A), resp. Dctf (Y, A)), and f and Rf ! b b map objects in Dc (Y, A) (resp. Dtf (Y, A), resp. Dctf (Y, A)) to objects in b b Dc (X, A) (resp. Dtf (X, A), resp. Dctf (X, A)). Proof. Using 7.5.6, 9.2.2 (ii) and the same argument as in the proof of 9.5.1, one can show that Rf∗ has finite cohomological dimension. Combined with 9.5.1 and 6.5.3, we see that Rf∗ maps objects in Dc (X, A) to objects b b in Dc (Y, A). By 6.5.6, it maps objects in Dtf (X, A) to objects in Dtf (Y, A). b b It follows that it maps objects in Dctf (X, A) to objects in Dctf (Y, A). The assertions for Rf! follows from 7.4.5, 7.4.7 (ii) and 7.8.1. The assertions for f ∗ are clear. To prove the assertions for Rf ! , the problem is local. We may i π assume that f can be factorized as the composite X → AdY → Y , where i is a closed immersion and π is the projection. We have Rf ! K ∼ = Ri! Rπ ! K = Ri! π ∗ K(d)[2d]. So it suffices to prove the assertions for Ri! . Let j : AdY − X ,→ AdY be the open immersion. We have a distinguished triangle i∗ Ri! K → K → Rj∗ j ∗ K → . ! The assertions for Ri follows from those for Rj∗ .

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Theorem 9.5.3. − (i) For any scheme X, the functor − ⊗L A − maps objects in Dc (X, A) × − b b b b Dc (X, A) (resp. Dtf (X, A) × Dtf (X, A), resp. Dctf (X, A) × Dctf (X, A)) to b b objects in Dc− (X, A) (resp. Dtf (X, A), resp. Dctf (X, A)). (ii) Suppose that S is a regular noetherian scheme of dimension ≤ 1. For any scheme X of finite type over S, the functor RH omA (−, −) maps b b objects in Dc− (X, A) × Dc+ (X, A) (resp. Dctf (X, A) × Dtf (X, A), resp. b b b Dctf (X, A) × Dctf (X, A)) to objects in Dc+ (X, A) (resp. Dtf (X, A), resp. b Dctf (X, A)). Proof. (i) Use 6.4.5, 6.4.6 and 6.5.4. (ii) By 6.5.3, to show that RH om maps objects in Dc− (X, A) × Dc+ (X, A) to objects in Dc+ (X, A), it suffices to show that for any constructible sheaves of A-modules F and G on X, E xtq (F , G ) are constructible for all q. First we prove that F has a filtration 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fk = F

such that Fi /Fi−1 are of the form j! F 0 for some immersion j : Y → X and some locally constant constructible sheaf of A-modules F 0 on Y . We use induction on dim X. This is clear if dim X ≤ 0. Suppose that dim X = d and the assertion holds for schemes of dimension < d. Let η1 , . . . , ηm be all the generic points of X. For each i, there exists an open neighborhood Ui of Sm ηi such that F |Ui is locally constant. Let U = i=1 Ui and let j : U ,→ X be the open immersion. F |U is locally constant, and j! (F |U ) is a subsheaf F . The quotient F /j! (F |U ) is supported on X − U which has dimension < n. We then apply the induction hypothesis to F /j! (F |U ). To prove our assertion, it suffices to show that E xtq (j! F 0 , G ) are constructible for any immersion j : Y → X, any locally constant constructible sheaf of Amodules F 0 on Y , and any constructible sheaf of A-modules G on X. We have RH om(j! F 0 , G ) ∼ = Rj∗ RH om(F 0 , Rj ! G ). By 9.5.2 applied to Rj∗ and Rj ! , we are reduced to proving that E xtq (F 0 , G 0 ) are constructible for any locally constant constructible sheaf of A-modules F 0 , and any constructible sheaf of A-modules G 0 on Y . The problem is local with respect to the etale topology on Y . So we may assume that F 0 is a constant constructible sheaf of A-modules, say associated to some finitely generated A-module M . Let · · · → L1 → L0 → 0

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be a resolution of M by free A-modules of finite rank. Then E xtq (F 0 , G 0 ) is the q-th cohomology sheaf of the complex 0 → H omA (L0 , G 0 ) → H omA (L1 , G 0 ) → · · · .

It is clear that it is constructible. b Next we show that RH om(−, −) maps objects in Dctf (X, A) × b b Dtf (X, A) to objects in Dtf (X, A). Let K be a bounded complex of flat constructible sheaves of A-modules. There exists a filtration 0 = K0 ⊂ K1 ⊂ · · · ⊂ Kk = K

such that Ki /Ki−1 are of the form j! K 0 for some immersion j : Y → X and some complex of locally constant flat constructible sheaves of A-modules K 0 . We have RH om(j! K 0 , L) ∼ = Rj∗ RH om(K 0 , Rj ! L). By 9.5.2, we are reduced to proving that RH om(K, L) has finite Tordimension for any bounded complex K of locally constant flat constructible b sheaves of A-modules and any L ∈ ob Dtf (X, A). The problem is local, we may assume that the components of K are constant sheaves. By 1.6.7, the components of K are constant sheaves of projective A-modules. We then have RH om(K, L) = H om· (K, L). As L has finite Tor-dimension, we may assume that L is a bounded complex of flat sheaves of A-modules. Then H om· (K, L) is a bounded complex of flat sheaves of A-modules. Hence RH om(K, L) has finite Tor-dimension. Using 9.3.1 and the same argument as above, one can prove the following propositions. Proposition 9.5.4. Let S be a noetherian scheme, X and Y two Sschemes of finite type, f : X → Y an S-morphism, and L ∈ ob Dcb (Y, A). Then there exists a dense open subset U of S such that the following conditions hold: (i) (Rf ! L)|XU is an object in Dcb (XU , A), where XU is the inverse image of U in X. (ii) The formation of Rf ! L commutes with any base change S 0 → U ⊂ S, or equivalently, the formation of RfU! (L|YU ) commutes with any base change S 0 → S, where YU is the inverse image of U in Y , and fU : XU → YU is the restriction of f . Proposition 9.5.5. Let S be a noetherian scheme, X an S-scheme of finite b type, K ∈ ob Dctf (X, A) and L ∈ ob Dcb (Y, A). Then there exists a dense open subset U of S such that the following conditions hold:

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(i) (RH omA (K, L))|XU is an object in Dcb (XU , A), where XU is the inverse image of U in X. (ii) The formation of RH omA (K, L) commutes with any base change S 0 → U ⊂ S, or equivalently, the formation of (RH omA (K, L))|XU commutes with any base change S 0 → S. 9.6

Biduality

([SGA 4 21 ] Th. finitude 4.) The main result of this section is the following: Theorem 9.6.1. Let S be a regular noetherian scheme of dimension ≤ 1 and let A be a noetherian ring such that A is an injective A-module, and nA = 0 for some integer n invertible on S. For any S-scheme f : X → S b of finite type, define KX = Rf ! A, and for any K ∈ ob Dctf (X, A), define b DK = RH omA (K, KX ). Then DK ∈ ob Dctf (X, A) and the morphism K → DDK = RH omA (DK, KX ) corresponding to the canonical morphism Ev L L ∼ K ⊗L A DK = DK ⊗A K = RH omA (K, KX ) ⊗A K → KX

is an isomorphism. Lemma 9.6.2. 9.6.1 holds if X = S. Proof. By definition, we have KS = A. To prove K ∼ = DDK, the problem is local with respect to the etale topology. If dim S = 0, any constructible sheaf of A-modules F on S is locally constant. So to prove the lemma in this case, we may assume that K is a bounded complex of finitely generated flat A-modules. By 1.6.7, all components of K are finitely generated projective A-modules. So we have DK = H om· (K, A),

DDK = H om· (H om· (K, A)).

b ∼ DDK. It is then clear that DK ∈ ob Dctf (S, A) and K = Suppose that S is a connected regular scheme of dimension 1. Given a constructible sheaf of A-modules F on S, let U be an open dense subset of S such that F |U is locally constant, and let j : U ,→ S and i : S − U → S

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be the immersions. By 8.3.10, we have D(j∗ j ∗ F ) = RH om(j∗ j ∗ F , A) ∼ = j∗ H om(j ∗ F , A), DD(j∗ j ∗ F ) ∼ = RH om(j∗ H om(j ∗ F , A), A)

∼ = j∗ H om(H om(j ∗ F , A), A). b It follows that D(j∗ j ∗ F ) ∈ ob Dctf (S, A) and j∗ j ∗ F ∼ = DD(j∗ j ∗ F ). Any constructible sheaf of A-modules on S supported on S − U is of the form i∗ G for a constructible sheaf of A-modules G on S − U . By 8.3.6 and the dim S = 0 case treated above, we have D(i∗ G ) = RH om(i∗ G , A) ∼ = i∗ RH om(G , Ri! A) ∼ = i∗ RH om(G , A(−1)[−2]) ∼ = i∗ H om· (G , A(−1)[−2]).

b Using this formula, one can verify that D(i∗ G ) ∈ ob Dctf (S, A) and i∗ G ∼ = DD(i∗ G ). Since the kernel and cokernel of the canonical morphism F → b j∗ j ∗ F are supported on S − U , it follows that D(F ) ∈ ob Dctf (S, A) and b ∼ F = DDF . We then conclude that for any K ∈ ob Dctf (S, A), we have b DK ∈ ob Dctf (S, A) and K ∼ = DDK.

Lemma 9.6.3. Under the assumption of 9.6.1, if f is proper, then we have Rf∗ K ∼ = Rf∗ DDK. Since f is proper, we have Rf! = Rf∗ , and hence Rf∗ DK = Rf∗ RH om(K, Rf ! A) ∼ = RH om(Rf∗ K, A) = DRf∗ K.

Proof.

We thus have a canonical isomorphism ∼ =

Rf∗ DK → DRf∗ K.

It follows that we have canonical isomorphisms ∼ =

Rf∗ DDK → DRf∗ DK,

By 9.6.2, we have

∼ =

DDRf∗ K → DRf∗ DK.

Rf∗ K ∼ = DDRf∗ K.

To prove Rf∗ K ∼ = Rf∗ DDK, it suffices to show that the following diagram commutes: Rf∗ K → Rf∗ DDK ∼ ↓∼ = ↓ = ∼ = DDRf∗ K → DRf∗ DK.

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Note that via 6.4.7, the composite Rf∗ K → DDRf∗ K → DRf∗ DK corresponds to the composite L L ∼ Rf∗ K ⊗L A Rf∗ DK → (Rf∗ K) ⊗A D(Rf∗ K) = D(Rf∗ K) ⊗A (Rf∗ K) → A,

and the composite Rf∗ K → Rf∗ DDK → DRf∗ DK corresponds to the composite Ev

L L Rf∗ K⊗L A Rf∗ DK → Rf∗ DDK⊗A Rf∗ DK → D(Rf∗ DK)⊗A (Rf∗ DK) → A.

So it suffices to prove the outer loop of the following diagram commutes: L L L (Rf∗ K)⊗L A D(Rf∗ K)← Rf∗ K⊗A Rf∗ DK → Rf∗ DDK⊗A Rf∗ DK →D(Rf∗ DK)⊗A (Rf∗ DK)

↓

(1)

↓

ok

Rf∗ (K⊗L A DK)

→

Rf∗ (DDK⊗L A DK)

D(Rf∗ K)⊗L A Rf∗ K

Rf∗ (DK⊗L A K)

(2)

↓ Rf∗ Rf ! A

↓

↓ Ev

↓

Rf∗ Rf ! A

=

↓

(3)

↓

A

=

A

=

A

↓ Ev

= A.

The commutativity of (1) and (3) is clear. The commutativity of (2) follows from the commutativity of the diagram L K ⊗L A DK → DDK ⊗A DK ↓ ↓ Ev Ev

DK ⊗L AK →

Rf ! A.

The leftmost square is of the same type as the rightmost square. Their commutativity follows from the commutativity of the outer loop of the following diagram: Rf∗ RH om(K, Rf ! A) ⊗L A Rf∗ K ↓ RH om(Rf∗ K, Rf∗ Rf ! A) ⊗L A Rf∗ K ↓ RH om(Rf∗ K, A) ⊗L A Rf∗ K

→ Rf∗ (RH om(K, Rf ! A) ⊗L A K) (4) ↓ → Rf∗ Rf ! A (5) ↓ → A.

In this diagram, the commutativity of (5) is clear. To prove the commutativity of (4), let I · be a complex of injective sheaves of A-modules

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quasi-isomorphic to Rf ! A, represent K by a complex of bounded complex of flat sheaves of A-modules, and let C · (K) be the Godement resolution. One can check directly that the following diagram commutes: f∗ H om· (C · (K), I · ) ⊗A f∗ C · (K) → f∗ (H om· (C · (K), I · ) ⊗A C · (K)) ↓ ↓ · · · · H om (f∗ C (K), f∗ I ) ⊗A f∗ C (K) → f∗ I · . b Proof of 9.6.1. By 9.5.2 and 9.5.3, we have DK ∈ ob Dctf (X, A). To ∼ prove K = DDK, the problem is local. By 5.9.10, we may assume that S is strictly local. Suppose that S is the spectrum of a separably closed field. We prove ∼ K = DDK by induction on dim X. The problem is local. We may assume that X is affine. Then X is a subscheme of a scheme projective over S. We are thus reduced to the case where X is proper over S. Let

K → DDK → ∆ → be a distinguished triangle. We need to show that ∆ is acyclic. Cover X by finitely many affine open subsets Xi so that we have immersions 1 m 1 X i → Am S . Let η be the generic point of AS and let πj : AS → AS (j = 1, . . . , m) be the projections. Applying the induction hypothesis to the generic fiber of πj |Xi , we see that ∆|Xi ∩π−1 (η) is acyclic for each pair j

(i, j). Since ∆ ∈ ob Dcb (X, A), there exists a nonempty open subset Vj ⊂ A1S such that ∆|Xi ∩π−1 (Vj ) is acyclic for each i by 5.9.8. We have j

Xi −

n [

j=1

(Xi ∩ πj−1 (Vj )) = Xi

\

(A1S − V1 ) ×S · · · ×S (A1S − Vn ) ,

which is finite over S. It follows that the cohomology sheaves of ∆ have finite support. On the other hand, we have Rf∗ ∆ = 0 by 9.6.3. So we have ∆ = 0. Suppose that S is a trait. Let s be the closed point of S. Let us prove that K ∼ = DDK by induction on dim Xs . Again we may assume that X is proper over S. Let K → DDK → ∆ → be a distinguished triangle. By the above discussion applied to the generic fiber Xη , the cohomology sheaves of ∆ are supported on Xs . Cover X by 0 finitely many affine open subsets Xi ⊂ Am S . Let S be the strict localiza1 0 tion of AS at the generic point s of the special fiber of A1S → S. Then

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S 0 is a strictly local trait. Applying the induction hypothesis to the morphisms Xi ×πj |Xi ,A1S S 0 → S 0 , we see that ∆|Xi ×π | ,A1 S 0 are acyclic. Then j Xi

S

there exist open neighborhoods Vi of s0 in A1S such that ∆|Xi ×π

1 Vi j |Xi ,AS

are

acyclic. Together with the fact that the cohomology sheaves of ∆ are supported on Xs , this implies that the cohomology sheaves are supported on finitely many points in Xs . Since Rf∗ ∆ = 0 by 9.6.3, we have ∆ = 0.

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Chapter 10

`-adic Cohomology

10.1

Adic Formalism

([SGA 5] V, VI, [Deligne (1980)] 1.1.) Let ` be a prime number, Z` = limn Z/`n+1 the ring of `-adic integers, Q` ←− its fraction field, E a finite extension of Q` , and R the integral closure of Z` in E. Then R is a complete discrete valuation ring. Fix a uniformizer λ of R. Then R/(λn+1 ) are finite and R/(λ) is a field of characteristic `. Throughout this chapter, we assume that ` is invertible in all schemes we consider. Let X be a noetherian scheme. A sheaf F of R-modules on X is called a λ-torsion sheaf if [ F = ker (λv : F → F ). v≥0

Consider the category of inverse systems of λ-torsion sheaves the form F = (Fn , un )n∈Z ,

un : Fn → Fn−1

so that Fn = 0 for all n ≤ n0 for some integer n0 . This is an abelian category. Given an integer r, we define F [r] to be the inverse system defined by (F [r])n = Fn+r with the transition morphisms given by un+r : (F [r])n → (F [r])n−1 . When r ≥ 0, we have a canonical morphism F [r] → F whose n-th component is un+1 · · · un+r : Fn+r → Fn . 531

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Similarly, we have a canonical morphism F → F [r] for any r ≤ 0. We say that a system F = (Fn ) satisfies the Mittag–Leffler condition (ML) if for any integer n, there exists an integer r ≥ n such that im(Fr → Fn ) = im(Ft → Fn ) for all t ≥ r. We say that it satisfies the Artin–Rees–Mittag–Leffler condition (ARML) if there exists an integer r such that im(F [r] → F ) = im(F [t] → F ) for all t ≥ r. We say that it is a null system if there exists an integer r ≥ 0 such that F [r] → F is 0. In the category of inverse systems of λ-torsion sheaves, the family of morphisms of the form F [r] → F is multiplicative. By 6.2.1, there exists a category, which we call the A-R category of inverse systems of λ-torsion sheaves, so that its objects are inverse systems of λ-torsion sheaves, and for any objects F and G , we have HomAR (F , G ) = lim Hom(F [r], G ). −→ r≥0

A morphism F → G in the A-R category is represented by a morphism F [r] → G of inverse systems for some r ≥ 0, and the kernel and cokernel of F [r] → G are the kernel and cokernel of F → G in the A-R category, respectively. The A-R category of inverse systems of λ-torsion sheaves is an abelian category. A system F is a zero object in the A-R category if and only if it is a null system. An inverse system F = (Fn , un )n∈Z of λ-torsion sheaves is called a λadic sheaf if Fn are constructible for all n, Fn = 0 for all n < 0, λn+1 Fn =0 for all n ≥ 0, and the morphisms un+1 : Fn+1 → Fn induce isomorphisms ∼ Fn Fn+1 /λn+1 Fn+1 = for all n ≥ 0. A λ-adic sheaf F is called lisse if Fn are locally constant for all n. An object F in the A-R category is called an A-R λ-adic sheaf if it is A-R isomorphic to a λ-adic sheaf, that is, we can find a λ-adic sheaf G and a morphism G [r] → F for some r ≥ 0 whose kernel and cokernel are null systems. Let G be a λ-adic sheaf and let F be an inverse system of λ-torsion sheaves such that λn+1 Fn = 0 for all n. Then we have Hom(G , F ) ∼ = HomAR (G , F ). So the category of λ-adic sheaves is equivalent to the category of A-R λ-adic sheaves.

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Examples. 1. Let M be a finitely generated R-module and let M/λn+1 M be the constant sheaf on X associated M/λn+1 M . Then M = (M/λn+1 M ) is a λ-adic sheaf. 2. Let un : µ`n+1 ,X → µ`n ,X be the morphism s 7→ sl . Then (µ`n+1 ,X ) is an `-adic sheaf. We denote it by Z` (1). Similarly, we have the `-adic sheaf Z` (m) = (Z/`n+1 (m)) for any integer m. 3. Let F be a constructible λ-torsion sheaf on X. Then (F /λn+1 F ) is a λ-adic sheaf. We have λm+1 F = 0 for a sufficiently large m and F /λn+1 F = F for all n ≥ m. The functor F → (F /λn+1 F ) identifies the category of λ-torsion sheaves with a full subcategory of the category of λ-adic sheaves. 4. If F = (Fn ) and G = (Gn ) are λ-adic sheaves, then F ⊗R G = (Fn ⊗R/(λn+1 ) Gn ) is a λ-adic sheaf. 5. If F = (Fn ) and G = (Gn ) are λ-adic sheaves, and each Fn is a locally free sheaf of R/(λn+1 )-modules of finite rank, then H omR (F , G ) = (H omR/λn+1 (Fn , Gn )) is a λ-adic sheaf. This follows from the fact that for each nonnegative integer n, we have an exact sequence λn+1

H omR/(λn+2 ) (Fn+1 , Gn+1 ) → H omR/(λn+2 ) (Fn+1 , Gn+1 ) → H omR/(λn+2 ) (Fn+1 , Gn ) → 0

and an isomorphism H omR/(λn+2 ) (Fn+1 , Gn ) ∼ = H omR/(λn+1 ) (Fn , Gn ). Proposition 10.1.1. Let F = (Fn ) be an inverse system of λ-torsion sheaves on a noetherian scheme X. Suppose that Fn are constructible for all n, λn+1 Fn = 0 for all n ≥ 0 and Fn = 0 for all n < 0. Then F is an A-R λ-adic sheaf if and only if the following two conditions hold: (a) F satisfies ARM L. (b) Let r ≥ 0 be an integer such that im (F [t] → F ) = im (F [r] → F ) for all t ≥ r. Such an integer exists by (a). Let F n = im (Fn+t → Fn )

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for any n and any t ≥ r. Then there exists an integer s ≥ 0 such that for any t ≥ s, the canonical morphisms Fn+t → Fn+s induce isomorphisms F n+t /λn+1 F n+t ∼ = F n+s /λn+1 F n+s

for all n ≥ 0 and all t ≥ s. If these conditions hold, then (F n+s /λn+1 F n+s ) is a λ-adic sheaf A-R isomorphic to F . Proof. (⇒) Let G = (Gn ) be a λ-adic sheaf A-R isomorphic to F = (Fn ). Since we have Hom(G , F ) = HomAR (G , F ), there exists a morphism G → F whose kernel and cokernel are null systems. So there exists an integer r ≥ 0 such that ker (Gn+r → Fn+r ) ⊂ ker (Gn+r → Gn ), im (Fn+r → Fn ) ⊂ im (Gn → Fn ) for all n ≥ 0. Since G is a λ-adic sheaf, we have Gn = im (Gm → Gn ) for all m ≥ n. It follows that im (Fn+r → Fn ) ⊂ im (Gn+t → Gn → Fn ) for all t ≥ r and hence im (Fn+r → Fn ) ⊂ im (Fn+t → Fn ) for all t ≥ r. This proves (a). Note that the morphisms Fn+t → Fn+r (t ≥ r) induce epimorphisms F n+t → F n+r . Let x be a section of F n+t such that its image in F n+r is 0. Since im (Fn+t+r → Fn+t ) ⊂ im (Gn+t → Fn+t ), locally with respect to the etale topology, we can lift x to a section y of Gn+t . As ker (Gn+r → Fn+r ) ⊂ ker (Gn+r → Gn ),

the image of y in Gn is 0, and hence y is a section of λn+1 Gn+t . Thus x is a section of λn+1 F n+t . This implies (b) by taking s = r. (⇐) Suppose conditions (a) and (b) hold. Let F = (F n ). The canonical morphism F [r] → F defines the A-R inverse of the inclusion F ,→ F . Hence F is A-R isomorphic to F . The canonical morphism F [s] → (F n+s /λn+1 F n+s ) defines the A-R inverse of the canonical morphism (F n+s /λn+1 F n+s ) → F . So F is A-R isomorphic to

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(F n+s /λn+1 F n+s ). We have n+2

(F n+1+s /λ

F n+1+s ) λn+1 (F n+1+s /λn+2 F n+1+s )

∼ = F n+1+s /λn+1 F n+1+s ∼ = F n+s /λn+1 F n+s .

So (F n+s /λn+1 F n+s ) is a λ-adic sheaf A-R isomorphic to F . Corollary 10.1.2. Let F = (Fn ) be an inverse system of λ-torsion sheaves on a noetherian scheme X. Suppose that Fn are constructible for all n, λn+1 F = 0 for all n ≥ 0 and Fn = 0 for all n < 0. (i) If F is A-R λ-adic, then so is f ∗ F for any morphism f : X 0 → X of schemes. (ii) Let {Ui → X}i be a finite etale covering of X. If each F |Ui is A-R λ-adic, then so is F . (iii) Let (Xi ) be a finite covering of X by locally closed subsets. If each F |Xi is A-R λ-adic, then so is F . Similarly, one can define the A-R category of inverse systems of λ-torsion R-modules, the λ-adic system of R-modules, and the A-R λ-adic system of R-modules. For example, a λ-adic system of R-modules is an inverse system F = (Fn ) such that Fn are finitely generated R-modules for all n, Fn = 0 for all n < 0, λn+1 Fn = 0 for all n ≥ 0, and the homomorphisms Fn+1 → Fn induce isomorphisms Fn+1 /λn+1 Fn+1 ∼ = Fn . The functor F = (Fn ) 7→ lim Fn ← − n is well-defined in the A-R category of inverse system of λ-torsion Rmodules since the canonical morphism F [r] → F induces an isomorphism limn Fn+r ∼ F for any r ≥ 0. = lim ←− ←−n n Lemma 10.1.3. Let

0 → (Fn ) → (Gn ) → (Hn ) → 0 be a short exact sequence of inverse system of R-modules. Suppose Fn is finite for all n. Then the sequence 0 → lim Fn → lim Gn → lim Hn → 0 ← − ← − ← − n n n is exact.

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Proof.

etale

Etale Cohomology Theory

It is clear that 0 → lim Fn → lim Gn → lim Hn ← − ← − ← − n n n

is exact. Let us prove that limn Gn → limn Hn is onto. Given an element ←− ←− (hn ) ∈ limn Hn , let En be the inverse images of hn in Gn . Then En are ←− finite. This implies that limn En 6= Ø. Let (en ) ∈ limn En . Then (en ) is an ←− ←− element in limn Gn that is mapped to (hn ). ←−

Proposition 10.1.4. (i) If F = (Fn ) is a λ-adic system of R-modules, then M = limn Fn is ←− a finitely generated R-module, and the projections M → Fn induce isomorphisms M/λn+1 M ∼ = Fn . (ii) The functor (Fn ) 7→ limn Fn defines an equivalence between the ←− category of A-R λ-adic systems of R-modules and the category of finitely generated R-modules. (iii) Let f : (Fn ) → (Gn ) be an A-R morphism of A-R λ-adic systems of R-modules. Then ker f and coker f are A-R λ-adic. Proof. (0) (0) (i) Let {x1 , . . . , xm } be a finite family of generators of F0 as an R(n) module. Choose xi ∈ Fn (i = 1, . . . , m) by induction on n so that they (n−1) are mapped to xi under the canonical homomorphisms Fn → Fn−1 . (n) (n) By Nakayama’s lemma, {x1 , . . . , xm } is a family of generators of Fn as an R-module. We have a family of epimorphisms (R/(λn+1 ))m → Fn ,

(a1 , . . . , am ) 7→

By 10.1.3, we have an epimorphism

m X

(n)

ai xi .

i=1

Rm ∼ (R/(λn+1 ))m → lim Fn . = lim ← − ← − n n It follows that M = limn Fn is a finitely generated R-module. For all ←− nonnegative integers m and n, we have an exact sequence λn+1

Fn+m → Fn+m → Fn → 0. Using 10.1.3, one verifies that the sequence λn+1

lim Fn+m → lim Fn+m → Fn → 0 ←− ←− m

m

n+1

is exact. So we have M/λ

M∼ = Fn .

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(ii) Let us show that the functor is fully faithful. Let F = (Fn ) and G = (Gn ) be λ-adic systems of R-modules, let M = limn Fn and N = limn Gn . ←− ←− We have HomAR (F, G) ∼ = Hom(F, G) = lim HomR/(λn+1 ) (Fn , Gn ) ← − n

∼ HomR/(λn+1 ) (M/λn+1 M, N/λn+1 N ). = lim ←− n

To prove that the functor is fully faithful, it suffices to show HomR (M, N ) ∼ HomR/(λn+1 ) (M/λn+1 M, N/λn+1 N ). = lim ← − n We can find an exact sequence Rv → Ru → M → 0. It gives rise to a commutative diagram HomR (M, N ) → HomR (Ru , N ) → HomR (Rv , N ) ↓ ↓ ↓ n+1 n+1 u limHom(M/λ M, limHom((R/(λ )) , limHom((R/(λn+1 ))v , − − − 0→← → ← → ← n n n n+1 n+1 N/λ N) N/λ N) N/λn+1 N).

0→

By the five lemma, to prove that the functor is fully faithful, it suffices to show that the last two vertical arrows are bijective. We are thus reduced to proving HomR (R, N ) ∼ HomR/(λn+1 ) (R/(λn+1 ), N/λn+1 N ), = lim ← − n or equivalently, N ∼ N/λn+1 N . This follows from (i). = lim ←−n For any finitely generated R-module M , (M/λn+1 M ) is a λ-adic system, and M ∼ M/λn+1 M . So the functor is essentially surjective. = lim ←−n (iii) We may assume (Fn ) and (Gn ) are λ-adic systems of R-modules. Then f is given by an element (fn ) in limn Hom(Fn , Gn ). Let M = limn Fn ←− ←− and N = limn Gn . Then M and N are finitely generated R-modules, ← − M/λn+1 M ∼ = Fn , and N/λn+1 N ∼ = Gn . Denote also by f the homomorphism M → N induced by fn : Mn → Nn by passing to limit. We have an exact sequence f

M → N → coker f → 0. It induces exact sequences M/λn+1 M → N/λn+1 N → coker f /λn+1 coker f → 0.

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These sequences can be identified with fn

Fn → Gn → coker f /λn+1 coker f → 0. So we have coker fn ∼ = coker f /λn+1 coker f . Hence coker f = (coker fn ) is λ-adic. We have canonical homomorphisms ker f /λn+1 ker f → ker fn . Let us show that their kernels and cokernels form null systems. This implies that ker f = (ker fn ) is A-R isomorphic to (ker f /λn+1 ker f ) and hence is AR λ-adic. By the Artin–Rees theorem ([Matsumura (1970)] (11.C) Theorem 15), there exists an integer r ≥ 0 such that im f ∩ λn N = λn−r (im f ∩ λr N ) for all n ≥ r. Suppose x + λn+1 M ∈ ker (M/λn+1 M → N/λn+1 N ), where x ∈ M . We have f (x) ∈ im f ∩ λn+1 N = λn+1−r (im f ∩ λr N ). n+1−r So f (x) = λ f (x0 ) for some x0 ∈ M . Then x − λn+1−r x0 ∈ ker f , and the image of x + λn+1 M in ker (M/λn+1−r M → N/λn+1−r N ) is equal to the image of (x − λn+1−r x0 ) + λn+1−r ker f ∈ ker f /λn+1−r ker f in ker (M/λn+1−r M → N/λn+1−r N ). It follows that the cokernels of ker f /λn+1 ker f → ker fn form a null system. By the Artin–Rees theorem, there exists an integer r ≥ 0 such that ker f ∩ λn M = λn−r (ker f ∩ λr M ) for all n ≥ r. Let x + λn+1 ker f be an element in the kernel of ker f /λn+1 ker f → ker fn , where x ∈ ker f . Then x ∈ ker f ∩ λn+1 M = λn+1−r (ker f ∩ λr M ). So x = λn+1−r x0 for some x0 ∈ ker f . The image of x + λn+1 ker f in ker f /λn+1−r ker f is 0. So the kernels of ker f /λn+1 ker f → ker fn form a null system. Let X be a scheme and let s → X be a geometric point of X. For any A-R λ-adic sheaf F = (Fn ) on X, the stalks (Fn,s ) form an A-R λ-adic system of R-modules. We define the stalk of F at s to be Fs = lim Fn,s . ← − n Proposition 10.1.5. Let F = (Fn ) be a λ-adic sheaf on a noetherian scheme X. There exists a dense open subset U of X such that Fn |U are locally constant for all n.

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Proof. For each n, the epimorphism λn+1 : Fn+1 → λn+1 Fn+1 vanishes on λFn+1 . Since Fn+1 /λFn+1 ∼ = F0 , it induces an epimorphism n+1 F0 → λ Fn+1 . Let Kn be the kernel of this epimorphism. Since F0 is constructible, the ascending chain · · · ⊂ Kn ⊂ Kn+1 ⊂ · · · in F0 is stationary by 5.8.5 (i). So we can find a dense open subset U of X such that F0 |U and Kn |U are locally constant for all n. Then F0 |U and λn+1 Fn+1 |U are locally constant for all n. Using induction on m and the fact that Fm+1 /λm+1 Fm+1 ∼ = Fm , one can show Fm+1 |U that are locally constant for all m. Remark 10.1.6. The same argument as in the proof of 10.1.5 shows that in the definition of an λ-adic sheaf F = (Fn ), we may replace the condition that Fn are constructible for all n by the condition that F0 is constructible. Proposition 10.1.7. (i) Let f : F → G be an A-R morphism between A-R λ-adic sheaves on a noetherian scheme X. Then ker f and coker f are A-R λ-adic. The category of A-R λ-adic sheaves form an abelian subcategory of the abelian A-R category of inverse system of λ-torsion sheaves. The category of λ-adic sheaves is an abelian category. (ii) Let f

g

F →G →H be a sequence of A-R λ-adic sheaves on a noetherian scheme X. It is exact if and only if for any x ∈ X, the sequence fx¯

gx¯

Fx¯ → Gx¯ → Hx¯ is exact. (iii) Let f

g

0→F →G →H →0 be a short exact sequence in the A-R category of inverse systems of λ-torsion sheaves on a noetherian scheme X. Suppose that Gn are constructible and there exists an integer t such that λn+t Gn = 0 for all n. If F and H are A-R λ-adic, so is G . Proof. (i) We may assume that F = (Fn ) and G = (Gn ) are λ-adic sheaves. Then f : F → G is given by an element (fn ) ∈ limn Hom(Fn , Gn ). To prove ←−

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that ker f and coker f are A-R λ-adic, we may use noetherian induction and 10.1.2 (iii) to reduce to the case where X is irreducible and prove that there exists a nonempty open subset U of X such that (ker fn |U ) and (coker fn |U ) are A-R λ-adic. By 10.1.5, there exists a nonempty open subset U of X such that Fn |U and Gn |U are locally constant. By 10.1.2 (ii) and 10.1.4 (iii), (ker fn |U ) and (coker fn |U ) are A-R λ-adic. (ii) We may assume that F = (Fn ), G = (Gn ) and H = (Hn ) are λ-adic sheaves. Then f and g are given by the elements (fn ) ∈ limn Hom(Fn , Gn ) and (gn ) ∈ limn Hom(Gn , Hn ), respectively. ←− ←− f g Suppose F → G → H is exact. Then gf = 0 in the A-R category. As HomAR (F , H ) ∼ = Hom(F , H ), we have (gn fn ) = 0. Hence gx¯ fx¯ = 0. The inclusion (im fn ) → (ker gn ) induces an isomorphism in the A-R category. Its cokernel is a null system. Hence the cokernel of (im fn,¯x ) → (ker gn,¯x ) is a null system. By 10.1.3, we have lim im fn,¯x ∼ ker gn,¯x , = lim ←− ←− n n lim im fn,¯x ∼ = im fx¯ , ←− n ∼ ker gx¯ , lim ker gn,¯x = ← − n fx¯ gx¯ So we have im fx¯ ∼ = ker gx¯ and the sequence Fx¯ → Gx¯ → Hx¯ is exact. fx¯ gx¯ Conversely suppose that Fx¯ → Gx¯ → Hx¯ is exact for all x ∈ X. To f g prove that F → G → H is exact, we may use noetherian induction to reduce to the case where X is irreducible and prove that there exists a nonempty open subset U of X such that F |U → G |U → H |U is exact. By 10.1.5, there exists a nonempty open subset U of X such that Fn |U , Gn |U fx¯

gx¯

and Hn |U are locally constant. Let x be a point in U . As Fx¯ → Gx¯ → Hx¯ is exact, (Fn,¯x ) → (Gn,¯x ) → (Hn,¯x ) is A-R exact by 10.1.4 (ii). This implies that F |U → G |U → H |U is A-R exact. (iii) Replacing G by G [−t + 1], we may assume that λn+1 Gn = 0 for all n. We may assume that F = (Fn ) is λ-adic. Then f : F → G is given by an element (fn ) ∈ limn Hom(Fn , Gn ). Let Hn = coker fn . By ←− our assumption, H = (Hn ) is an A-R λ-adic sheaf. So there exist integers r, s ≥ 0 such that the conditions (a) and (b) in 10.1.1 hold for H . Chasing

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the following commutative diagram Fn+t → Gn+t → Hn+t → 0 ↓ ↓ ↓ Fn+r → Gn+r → Hn+r → 0 ↓ ↓ ↓ Fn → Gn → Hn → 0, one gets im (G [t] → G ) = im (G [r] → G ) for all t ≥ r. So condition (a) in 10.1.1 holds for G . Replacing G by G = (G n ), where G n = im (Gn+t → Gn ) for all t ≥ r and all n, we may assume that Gn+1 → Gn are surjective. By our assumption, ker (Fn → Gn ) forms a null system. We may choose the integer s such that condition (b) in 10.1.1 holds for H and ker (Fn → Gn ) ⊂ ker (Fn → Fn−s ) for all n ≥ s. Let us prove Gn+t /λn+1 Gn+t ∼ = Gn+2s /λn+1 Gn+2s for all n and all t ≥ 2s. Thus the condition (b) in 10.1.1 holds for G , and G is A-R λ-adic. Let y be a section in ker (Gn+t → Gn+2s ). Then the image of y in Hn+t lies in ker (Hn+t → Hn+2s ). Since H satisfies the conditions in 10.1.1, we have ker (Hn+t → Hn+2s ) ⊂ λn+s+1 Hn+t . So locally for the etale topology, there exists a section y 0 of Gn+t such that y − λn+s+1 y 0 = fn+t (x) for some section x of Fn+t . Since y is a section of ker (Gn+t → Gn+2s ) and λn+s+1 y 0 is a section of ker (Gn+t → Gn+s ), the image of x in Fn+s is a section of ker (Fn+s → Gn+s ). By our choice of s, we have ker (Fn+s → Gn+s ) ⊂ ker (Fn+s → Fn ). So x is a section of ker (Fn+t → Fn ) = λn+1 Fn+t . It follows that y = λn+s+1 y 0 + fn+t (x) is a section of λn+1 Gn+t . This implies that Gn+t /λn+1 Gn+t ∼ = Gn+2s /λn+1 Gn+2s .

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Proposition 10.1.8. Let X be a noetherian scheme, F an A-R λ-adic sheaf, and F (0) ⊂ F (1) ⊂ · · · an ascending chain of A-R λ-adic subsheaves in F . We have F (m) = F (m+1) = · · · for sufficiently large m. Proof. Using noetherian induction, to show the chain is stationary, we may assume that X is irreducible and prove that there exists a nonempty open subset U of X such that the chain F (0) |U ⊂ F (1) |U ⊂ · · · is stationary. Let η be the generic point of X. We have a chain of finitely generated R-modules (0)

Fη¯

(1)

⊂ Fη¯

⊂ ··· .

This chain is stationary. Let m ≥ 0 be an integer such that (m)

Fη¯

(m+1)

= Fη¯

= ··· .

By 10.1.5, there exists a nonempty open subset U of X such that (F /F (m) )|U is A-R isomorphic to a lisse λ-adic sheaf on U . For any (m) (m) x ∈ U , the specialization homomorphism Fx¯ /Fx¯ → Fη¯ /Fη¯ is an iso(m) (i) (m) (i) morphism. So the specialization homomorphism Fx¯ /Fx¯ → Fη¯ /Fη¯ (i)

(m)

(i)

(m)

is injective for any i ≥ m. We have Fη¯ /Fη¯ = 0. So Fx¯ /Fx¯ = 0 for any i ≥ m and any x ∈ U . It follows that (F (i) /F (m) )|U = 0 for any i ≥ m, and hence the chain is stationary when restricted to U . Corollary 10.1.9. For any λ-adic sheaf F on a noetherian scheme X, there exist λ-adic sheaves F 0 and F 00 and a short exact sequence 0 → F 0 → F → F 00 → 0 in the A-R category such that the following conditions hold: (a) F 0 = (Fn0 ) is a torsion sheaf, that is, there exists an integer m ≥ 0 0 such that Fn0 = Fm for all n ≥ m. 00 00 (b) F = (Fn ) is torsion free, that is, the morphism λ : F 00 → F 00 is A-R injective. Fn00 are flat sheaves of R/(λn+1 )-modules for all n.

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Proof. Let K (n) be the kernel of λn : F → F . Then K (n) form an ascending chain of A-R λ-adic subsheaves of F . By 10.1.8, there exists an integer m ≥ 0 such that K

0

(m)

=K

(m+1)

00

= ··· .

Let F and F be the λ-adic sheaves A-R isomorphic to K (m) and F /K (m) , respectively. Then λm : F 0 → F 0 is 0 in the A-R category. So λm

there exists an integer r ≥ 0 such that the composite F 0 [r] → F 0 → F 0 is 0 0. As Fn+r → Fn0 is onto for all n, λm : F 0 → F 0 is 0, that is, λm Fn0 = 0 for all n. It follows that for any n ≥ m, we have ∼ F0 . F 0 = F 0 /λm+1 F 0 = n

n

n

m

So the condition (a) holds. The kernel of the composite λ

is K

(m+1)

F → F → F /K

(m)

. So we have an A-R monomorphism λ : F /K

(m)

(m+1)

(m+1)

→ F /K

(m)

.

(m)

But K = K . So λ : F /K → F /K (m) is A-R injective. 00 00 Hence λ : F → F is A-R injective. For any x ∈ X, the homomorphism λ : Fx¯00 → Fx¯00 is injective. So Fx¯00 is a finitely generated torsion free Rmodule. As R is a discrete valuation ring, this implies that Fx¯00 is free of 00 00 n+1 finite rank. Then Fn,¯ Fx¯00 are free of finite rank for all n. x = Fx ¯ /λ 00 n+1 Hence Fn are flat sheaves of R/(λ )-modules. In the following (10.1.10–12), we fix a local noetherian ring A and a proper ideal I of A. Let A0 = A/I. For any A-module M , let M0 = M ⊗A A0 . For all free A-modules M and N of finite ranks and any homomorphism φ0 : M0 → N0 , there exists a homomorphism φ : M → N such that φ0 = φ ⊗ idA0 . Moreover, φ identifies M with a direct factor of N if and only if φ0 identifies M0 with a direct factor of N0 . The “only if” part is clear. To prove the “if” part, let ψ0 : N0 → M0 be a homomorphism such that ψ0 φ0 = id, and let ψ : N → M be any lift of ψ0 . Then det (ψφ) ≡ det (ψ0 φ0 ) ≡ 1 mod I.

It follows that det (ψφ) is a unit, and hence ψφ is invertible. Let δ : M → M be the inverse of ψφ. Then (δψ)φ = id. So φ identifies M with a direct factor of N . Lemma 10.1.10. Let M0· be a bounded acyclic complex of free A0 -modules of finite ranks. There exists a bounded acyclic complex M · of free A-modules of finite ranks such that M · ⊗A A0 ∼ = M0· .

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Without loss of generality, suppose that M0· is of the form d0

dn−1

0 0 → M00 →0 M01 → · · · → M0n → 0.

Let Z0i = ker di0 . We have short exact sequences 0 → Z0n−1 → M0n−1 → M0n → 0, 0 → Z0n−2 → M0n−2 → Z0n−1 → 0, .. . 0 → M00 → M01 → Z02 → 0. Choose free A-modules of finite rank M i such that M i ⊗A A0 ∼ = M0i for all i. n−1 n Lift the homomorphism M0 → M0 to a homomorphism M n−1 → M n . By Nakayama’s lemma, the lifting is surjective. Let Z n−1 be its kernel. The short exact sequence 0 → Z n−1 → M n−1 → M n → 0 splits. So Z n−1 is free of finite rank and Z n−1 ⊗A A0 ∼ = Z0n−1 . Lift the n−2 n−1 n−2 n−1 homomorphism M0 → Z0 to M → Z and let Z n−2 be its kernel. Then we have a split short exact sequence 0 → Z n−2 → M n−2 → Z n−1 → 0, Z n−2 is free of finite rank, and Z n−2 ⊗A A0 ∼ = Z0n−2 . In this way, we get split short exact sequences 0 → Z n−1 → M n−1 → M n → 0, 0 → Z n−2 → M n−2 → Z n−1 → 0, .. . 0 → Z 1 → M 1 → Z 2 → 0. Finally we take M 0 = Z 1 . Then the complex 0 → M0 → M1 → · · · → Mn → 0 has the required property.

Lemma 10.1.11. Let φ : M · → N · be a morphism of bounded complexes of free A-modules of finite ranks, and let φ00 : M0· → N0· be a morphism of complexes homotopic to φ0 : M0· → N0· . Then there exists a morphism of complexes φ0 : M · → N · such that φ00 = φ0 ⊗ idA0 and φ0 is homotopic to φ.

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Proof. is,

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Let k0 : M0· → N0· [−1] be a homotopy between φ0 and φ00 , that dk0 + k0 d = φ0 − φ00 .

Let k : M · → N · [−1] be a lift of k0 and define φ0 = φ − dk − kd. Then dφ0 = φ0 d. So φ0 : M · → N · is a morphism of complex. It has the required property. Lemma 10.1.12. Let M · (resp. N0· ) be a bounded complex of free Amodules (resp. free A0 -modules) of finite ranks and let φ0 : M0· → N0· be a quasi-isomorphism. Then there exists a bounded complex N · of free Amodules of finite ranks and a quasi-isomorphism φ : M · → N · such that N · ⊗A A0 ∼ = N0· and φ ⊗ idA0 = φ0 . Proof. Let C0· be the mapping cone of id : N0· → N0· . It is a bounded acyclic complex of free A0 -modules of finite rank. By 10.1.10, we can lift it to a bounded acyclic complex C · of free A-modules of finite rank. Replacing M · by M · ⊕ C · [−1] and replacing φ0 by the morphism M0· ⊕ C0· [−1] → N0· defined by the homomorphisms M0p ⊕ N0p ⊕ N0p−1 → N0p ,

(x, y, z) 7→ φ0 (x) + y,

we may assume φ0 is surjective. Let K0· = ker φ0 . Then K0· is a bounded acyclic complex of free A0 -modules. By 10.1.10, it can be lifted to a bounded acyclic complex K · of free A-modules. The complex K0 is split. By 6.4.11, K0 is homotopic to the zero complex. So the inclusion i0 : K0· ,→ M0· is homotopic to 0. By 10.1.11, there exists a morphism of complexes i : K · → M · homotopic to 0 such that i ⊗ idA0 = i0 . Since i0 identifies each K0n with a direct factor M0n , i identifies each K n with a direct factor M n . Set N · = M · /K · . Then N · has the required property. Lemma 10.1.13. Let A be a ring, L· and N · complexes of A-modules, and π : N · → L· a quasi-isomorphism. Then for any bounded above complex of projective A-modules M · , and any morphism of complexes φ : M · → L· , there exists a morphism of complexes ψ : M · → N · such that πψ is homotopic to φ. Proof. Let C · be the mapping cone of id : L· → L· . Then C · is homotopic to 0. Hence N · is homotopic to N · ⊕ C · [−1]. Let π 0 : N · ⊕ C · [−1] → L· ,

π 00 : N · ⊕ C · [−1] → N · ,

i : N · → N · ⊕ C · [−1]

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be the morphisms of complexes defined by the homomorphisms N p ⊕ Lp ⊕ Lp−1 → Lp , p

p

N ⊕L ⊕L p

p

p−1 p

p

→N ,

N →N ⊕L ⊕L

p−1

,

(x, y, z) 7→ π(x) + y, (x, y, z) 7→ x, x 7→ (x, 0, 0),

respectively. Then we have π 0 i = π, and iπ 00 is homotopic to idN · ⊕C · [−1] . So ππ 00 = π 0 iπ 00 is homotopic to π 0 . Suppose we can find a morphism of complexes ψ 0 : M · → N · ⊕ C · [−1] such that π 0 ψ 0 is homotopic to φ. Taking ψ = π 00 ψ 0 , then πψ is homotopic to φ. Replacing N · by N · ⊕ C · [−1] and replacing π by π 0 , we are reduced to the case where π is surjective. Let us prove that under the assumption that π is surjective, we can find a morphism of complexes ψ : M · → N · such that πψ = φ. Let K · = ker π. It is acyclic. Define ψ q : M q → N q to be 0 for those q such that M i = 0 for all i ≥ q. Suppose we have defined ψ q for all q > r. Since M r is projective and π r : N r → Lr is onto, there exists a homomorphism ψ1r : M r → N r such that π r ψ1r = φr . We have π r+1 (drN · ψ1r − ψ r+1 drM · ) = drL· π r ψ1r − φr+1 drM · = drL· φr − φr+1 drM · = 0. So drN · ψ1r − ψ r+1 drM · maps M r to K r+1 . We have r r r+1 r r r r+2 r+1 r dr+1 dM · ) = dr+1 dM · dM · = 0. N · (dN · ψ1 − ψ N · dN · ψ1 − ψ

So drN · ψ1r − ψ r+1 drM · maps M r to ker (d : K r+1 → K r+2 ). Since K · is acyclic and M r is projective, there exists a homomorphism ψ2r : M r → K r such that drN · ψ2r = drN · ψ1r − ψ r+1 drM · . We then define ψ r = ψ1r − ψ2r .

Lemma 10.1.14. Let A be a noetherian local ring. (i) Let M · be a bounded above complex of A-modules such that H i (M · ) are finitely generated A-modules. Then there exists a quasi-isomorphism s : M 0· → M · such that M 0· is a bounded above complex of free A-modules of finite ranks. (ii) Let M · be a bounded complex of flat A-modules such that H i (M · ) are finitely generated A-modules and that M i = 0 for i 6∈ [a, b] for some integers a and b. Then there exists a quasi-isomorphism s : M 0· → M · such that M · is a bounded complex of free A-modules of finite ranks, and M 0i = 0 for i 6∈ [a, b].

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Proof. Since A is a noetherian local ring, a finitely generated A-module is free if and only if it is flat. Using the same method as the proof of 6.4.5, one can prove (i), and under the assumption of (ii), one can construct a quasi-isomorphism F · → M · such that F i are finitely generated flat Amodules, and F i = 0 for all i > b. Since M · is a bounded above complex of flat A-modules, by 6.4.4, we have H i (F · ⊗A N ) ∼ = H i (M · ⊗A N ) for any A-module N . Since M i = 0 for all i < a, we have H i (F · ⊗A N ) = 0 for all i < a. Note that · · · → F a−1 → F a → 0 is a flat resolution of coker (F a−1 → F a ). It follows that a−1 TorA → F a ), N ) ∼ = H a−i (F · ⊗A N ) = 0 i (coker (F

for all i > 0. So coker (F a−1 → F a ) is flat. Let M 0· be the complex 0 → coker (F a−1 → F a ) → F a+1 → · · · → F b → 0. Then the quasi-isomorphism F · → M · induces a quasi-isomorphism M 0· → M · , and M 0· is a complex of free A-modules of finite ranks. Proposition 10.1.15. Let (Kn· )n∈Z be an inverse system of complexes of R-modules and let a ≤ b be integers. Suppose the following conditions hold: (a) Kn· = 0 for n < 0, and Kni = 0 for i 6∈ [a, b]. (b) λn+1 Kni = 0 and Kni are flat R/(λn+1 )-modules for all n ≥ 0 and i. (c) H i (Kn· ) are finite. (d) The morphisms · Kn+1 ⊗R/(λn+2 ) R/(λn+1 ) → Kn· · induced by Kn+1 → Kn· are quasi-isomorphism for all n ≥ 0. Then we can find a bounded complex K · of free R-modules of finite ranks and quasi-isomorphisms K · /λn+1 K · → Kn· such that the diagrams · K · /λn+2 K · → Kn+1 ↓ ↓ · n+1 · K /λ K → Kn·

commute up to homotopy. The inverse systems (H i (Kn· ))n∈Z are A-R λadic, and H i (K · ) ∼ H i (Kn· ). = lim ←−n

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Proof. By 10.1.14, we can find quasi-isomorphisms Kn0· → Kn· (n ≥ 0) such that Kn0· are complexes of free R/(λn+1 )-modules of finite ranks and Kn0i = 0 for i 6∈ [a, b]. Let K000· = K00· . By 10.1.13, we can find a morphism K10· ⊗R/(λ2 ) R/(λ) → K000·

such that the following diagram commutes up to homotopy: K10· ⊗R/(λ2 ) R/(λ) → K1· ⊗R/(λ2 ) R/(λ) ↓ ↓ K000· → K0· .

Note that K10· ⊗R/(λ2 ) R/(λ) → K000· is necessarily a quasi-isomorphism. By 10.1.12, we can find a bounded complex K100· of free R/(λ2 )-modules of finite rank and a quasi-isomorphism K10· → K100· such that K100· ⊗R/(λ2 ) R/(λ) ∼ = K000·

and that the morphism K10· ⊗R/(λ2 ) R/(λ) → K000· is induced by the morphism K10· → K100· . By the dual version of 6.2.7 for complexes of projective objects, the quasi-isomorphism K10· → K100· is homotopically invertible. Let K100· → K1· be the composite of a homotopic inverse of K10· → K100· with the quasi-isomorphism K10· → K1· . Then K100· → K1· is a quasi-isomorphism, and the following diagram commutes up to homotopy: K100· → K1· ↓ ↓ K000· → K0· . Similarly, we can find a quasi-morphism K20· ⊗R/(λ3 ) R/(λ2 ) → K100· such that the diagram K20· ⊗R/(λ3 ) R/(λ2 ) → K2· ⊗R/(λ3 ) R/(λ2 ) ↓ ↓ K100· → K1·

commutes up to homotopy, and we can find a bounded complex K200· of free R/(λ3 )-modules of finite ranks and a quasi-isomorphism K20· → K200· such that K200· ⊗R/(λ3 ) R/(λ2 ) ∼ = K100· and that the morphism K20· ⊗R/(λ3 ) R/(λ2 ) → K100· is induced by the morphism K20· → K200· . Let K200· → K2· be the composite of a homotopic inverse

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of K20· → K200· with the quasi-isomorphism K20· → K2· . Then K200· → K2· is a quasi-isomorphism, and the following diagram commutes up to homotopy: K200· → K2· ↓ ↓ K100· → K1· .

In this way, we can construct a family of bounded complexes of free R/(λn+1 )-modules of finite ranks Kn00· and quasi-isomorphisms Kn00· → Kn such that 00· Kn+1 ⊗R/(λn+2 ) R/(λn+1 ) ∼ = Kn00·

and the diagrams 00· · Kn+1 → Kn+1 ↓ ↓ Kn00· → Kn·

commute up to homotopy. Since K000i = 0 for i 6∈ [a, b], we have Kn00i = 0 for i 6∈ [a, b] by Nakayama’s lemma. Set K · = limn Kn00· . Then K · has the ←− required property. By 10.1.4 (iii), (H i (Kn00· ))n∈Z are A-R λ-adic, and by 10.1.3, we have H i (K · ) ∼ H i (K 00· ). It follows that (H i (Kn· ))n∈Z are = lim ←−n i n· i · ∼ A-R λ-adic, and H (K ) = limn H (Kn ). ←−

Proposition 10.1.16. Let X be a noetherian scheme, and let K = (Kn , un )n≥0 be a family consisting of bounded above complexes of sheaves of R/(λn+1 )-modules Kn on X and isomorphisms n+1 ∼ un : Kn+1 ⊗L ) = Kn R/(λn+2 ) R/(λ

in D− (X, R/(λn+1 )). If H i (K0 ) are constructible sheaves of R/(λ)modules and are nonzero for only finitely many i, then Kn ∈ b ob Dctf (X, R/(λn+1 )) for all n, and the inverse systems (H i (Kn ))n∈Z are A-R λ-adic. Proof. Represent each Kn by a bounded above complex of sheaves of flat R/(λn+1 )-modules. We have n L L ∼ Kn ⊗R/(λn+1 ) R/(λ) ∼ = Kn ⊗L R/(λn+1 ) R/(λ )⊗R/(λn ) · · ·⊗R/(λ2 ) R/(λ) = K0 .

On the other hand, we have λi Kn /λi+1 Kn ∼ = Kn ⊗R/(λn+1 ) (λi )/(λi+1 ) ∼ = Kn ⊗R/(λn+1 ) R/(λ) ∼ = K0 . So each Kn has a finite filtration 0 = λn+1 Kn ⊂ λn Kn ⊂ · · · ⊂ λKn ⊂ Kn

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such that the successive quotients λi Kn /λi+1 Kn are isomorphic to K0 . As H i (K0 ) are constructible, so are H i (Kn ). Suppose that H i (K0 ) = 0 for i 6∈ [a, b]. Then we have H i (Kn ) = 0 for i 6∈ [a, b]. The complex · · · → Kna−1 → Kna → 0 is a resolution of coker (Kna−1 → Kna ) by flat sheaves of R/(λn+1 )-modules. So R/(λn+1 )

Tori

We have H

(coker (Kna−1 → Kna ), R/(λ)) ∼ = H a−i (Kn ⊗R/(λn+1 ) R/(λ)) ∼ = H a−i (K0 ).

a−i

(K0 ) = 0 for all i > 0. So R/(λn+1 )

Tori

(coker (Kna−1 → Kna ), R/(λ)) = 0

for all i > 0. By 1.2.8, coker (Kna−1 → Kna ) is a flat sheaf of R/(λn+1 )modules. The complex Kn is quasi-isomorphic to 0 → coker (Kna−1 → Kna ) → Kna+1 → · · · , and the latter is a bounded complex of flat sheaves of R/(λn+1 )-modules. b So Kn has finite Tor-dimension. It follows that Kn ∈ ob Dctf (X, R/(λn+1 )). i To prove that the inverse systems (H (Kn ))n∈Z are A-R λ-adic, we use noetherian induction, and we may assume that X is irreducible and prove that there exists a nonempty open subset U of X such that (H i (Kn )|U )n∈Z are A-R λ-adic. There exists a nonempty open subset U such that H i (K0 )|U are locally constant for all i. This implies that H i (Kn )|U are locally constant for all i and n. Let x be an arbitrary point in U . To prove our assertion, it suffices to show that (H i (Kn,¯x ))n∈Z are A-R λ-adic. By 10.1.14, each Kn,¯x is isomorphic in the derived category to a complex L·n of free R/(λn+1 )-modules of finite ranks with Lvn = 0 for v 6∈ [a, b]. The isomorphism n+1 ∼ Kn+1 ⊗L ) = Kn R/(λn+2 ) R/(λ

induces an isomorphism L·n+1 ⊗R/(λn+2 ) R/(λn+1 ) ∼ = L·n in the derived category. By 10.1.13, the latter isomorphism is induced by a quasi-isomorphism L·n+1 ⊗R/(λn+2 ) R/(λn+1 ) ∼ = L·n of complexes. Applying 10.1.15 to the inverse system of complexes (L·n )n∈Z , we see that (H i (L·n ))n∈Z are A-R λ-adic. So (H i (Kn,¯x ))n∈Z are A-R λ-adic.

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Let X be a noetherian scheme. Define a category Dcb (X, R) as follows: Objects in Dcb (X, R) are families K = (Kn , un )n≥0 consisting of objects Kn in D− (X, R/(λn+1 )) and isomorphisms n+1 ∼ un : Kn+1 ⊗L ) = Kn R/(λn+2 ) R/(λ

in D− (X, R/(λn+1 )) such that K0 ∈ ob Dcb (X, R/(λ)). Let K = (Kn , un )n≥0 and K 0 = (Kn0 , u0n )n≥0 be two objects in Dcb (X, R). A morphism f : K → K 0 in Dcb (X, R) is a family (fn ) of morphisms fn : Kn → Kn in D(X, R/(λn+1 )) such that fn un = u0n (fn+1 ⊗L R/(λn+2 ) idR/(λn+1 ) ). If F = (Fn ) is a torsion free λ-adic sheaf, then F defines an object in Dcb (X, R). Proposition 10.1.17. (i) Let f : X → Y be a morphism between noetherian schemes. For any K = (Kn ) ∈ ob Dcb (Y, R), we have f ∗ K = (f ∗ Kn ) ∈ ob Dcb (X, R). (ii) Let f : X → Y be an S-compactifiable morphism between noetherian schemes. For any K = (Kn ) ∈ ob Dcb (X, R), we have Rf! K = (Rf! Kn ) ∈ ob Dcb (Y, R). (iii) Let S be a noetherian regular scheme of dimension ≤ 1, X and Y two S-schemes of finite type, f : X → Y an S-compactifiable morphism, K = (Kn ) and L = (Ln ) objects in Dcb (X, R), and M = (Mn ) an object in Dcb (Y, R). Define Rf∗ K = (Rf∗ Kn ), f ∗ M = (f ∗ Mn ), Rf! K = (Rf! Kn ), Rf ! M = (Rf ! Mn ), L K ⊗L R L = (Kn ⊗R/(λn+1 ) Ln ),

RH om(K, L) = (RH om(Kn , Ln )).

Then Rf∗ K and Rf! K are objects in Dcb (Y, R), while f ∗ M , Rf ! M , K ⊗L RL and RH om(K, L) are objects in Dcb (X, R). (iv) Let S be a trait, X an S-scheme of finite type, η the generic point of S, s the closed point of S, K = (Kn ) ∈ ob Dcb (Xη , R), and L = (Ln ) ∈ ob Dcb (X, R). Define RΨη (K) = (RΨη (Kn )),

RΦ(L) = (RΦ(Ln )).

Then RΨη (K) and RΦ(L) are objects in Dcb (Xs¯, R).

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(v) Let S be a noetherian regular scheme of dimension ≤ 1 and let f : X → S be an S-scheme of finite type. For any K = (Kn ) ∈ ob Dcb (X, R), define ! n+1 DK = RH om Kn , Rf (R/(λ )) .

Then the canonical morphism K → DDK is an isomorphism.

Proof. We prove (iii) and leave the rest to the reader. By 10.1.16, we b have Kn ∈ ob Dctf (X, R/(λn+1 )). By 7.4.7 (ii) and 7.8.1, we have Rf! Kn ∈ b n+1 ob Dctf (Y, R/(λ )). By 7.4.7 (iii), we have L n+1 ∼ Rf! Kn+1 ⊗ ) = Rf! (Kn+1 ⊗L n+2 R/(λn+1 )) ∼ = Rf! Kn . n+2 R/(λ R/(λ

)

R/(λ

)

So Rf! K ∈ ob Dcb (Y, R). Similarly, using 6.5.7 and 9.5.2, one can show Rf∗ K ∈ ob Dcb (Y, R). One can easily show that f ∗ M and K ⊗L R L are b objects in Dcb (X, R). By 9.5.2, we have Rf ! Mn ∈ ob Dctf (X, R/(λn+1 )). To prove Rf ! M ∈ ob Dcb (X, R), it suffices to show n+1 ∼ n+1 Rf ! Mn+1 ⊗L ) = Rf ! (Mn+1 ⊗L )). R/(λn+2 ) R/(λ R/(λn+2 ) R/(λ

Let A be a noetherian ring and let B be a noetherian A-algebra. Let us b prove more generally that for any M ∈ ob Dctf (Y, A), we have an isomorphism Rf ! M ⊗L B ∼ = Rf ! (M ⊗L B) A

A

in D(X, B). First we construct a morphism ! L Rf ! M ⊗L A B → Rf (M ⊗A B)

in D(X, B). Let ρ : D(X, B) → D(X, A) and ρ : D(Y, B) → D(Y, A) be the functors in 8.4.12. It suffices to construct a morphism Rf ! M → ρRf ! (M ⊗L A B) in D(X, A). By 8.4.13, we have a canonical isomorphism ρRf ! (M ⊗L B) ∼ = Rf ! ρ(M ⊗L B). A

A

So we need to construct a morphism Rf ! M → Rf ! ρ(M ⊗L A B). We define it to be the morphism induced by the canonical morphism M → ρ(M ⊗L A B).

! L To prove that the morphism Rf ! M ⊗L A B → Rf (M ⊗A B) is an isomorphism is a local problem. So we may assume that f is a composite i

π

X → AdY → Y,

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so that i is a closed immersion and π is the projection. We have L ∼ ! ∗ Rf ! M ⊗L A B = Ri π M (d)[2d] ⊗A B, L L ∼ ! ∗ ∼ ! ∗ Rf ! (M ⊗L A B) = Ri π (M ⊗A B)(d)[2d] = Ri (π M (d)[2d] ⊗A B).

To prove our assertion, it suffices to show that L ∼ ! Ri! M ⊗L A B = Ri (M ⊗A B) b for any M ∈ ob Dctf (AdY , A). Let j : AnY − i(X) → AnY be the open immersion. We have a morphism of distinguished triangles L ∗ L i∗ Ri! M ⊗L A B → M ⊗A B → Rj∗ j M ⊗A B → ↓ ↓ ↓ L ∗ i∗ Ri! (M ⊗L B) → M ⊗ B → Rj j (M ⊗L ∗ A A A B) → .

The second and the third vertical arrows are isomorphisms. So the first arrow is also an isomorphism. Our assertion follows. b By 9.5.3 (ii), we have RH om(Kn , Ln ) ∈ ob Dctf (X, R/(λn+1 )). To b prove RH om(K, L) ∈ ob Dc (X, R), we need to show n+1 RH om(Kn+1 , Ln+1 ) ⊗L ) R/(λn+2 ) R/(λ

n+1 n+1 ∼ ), Ln+1 ⊗L )). = RH om(Kn+1 ⊗L R/(λn+2 ) R/(λ R/(λn+2 ) R/(λ

Let A be a noetherian ring and let B be a noetherian A-algebra. Let b us prove more generally that for any K, L ∈ ob Dctf (X, A), we have an isomorphism L L ∼ RH om(K, L) ⊗L A B = RH om(K ⊗A B, L ⊗A B).

We have a canonical morphism Ev

RH om(K, L) ⊗L A K → L. It induces a morphism Ev

L L L (RH om(K, L) ⊗L A B) ⊗B (K ⊗A B) → L ⊗A B.

By 6.4.7, this gives rise to a morphism L L RH om(K, L) ⊗L A B → RH om(K ⊗A B, L ⊗A B).

To prove that it is an isomorphism, we may assume K = j! F such that j : Y → X is an immersion, and F is a locally constant constructible sheaf of A-modules on Y . This is because K can be represented by a bounded complex of constructible sheaves of A-modules, and any constructible sheaf

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of A-modules has a finite filtration such that the successive quotients are of the form j! F . We have ! L ∼ RH om(j! F , L) ⊗L A B = Rj∗ RH om(F , Rj L) ⊗A B ∼ = Rj∗ (RH om(F , Rj ! L) ⊗L B). A

L L L ∼ RH om(j! F ⊗L A B, L ⊗A B) = RH om(j! (F ⊗A B), L ⊗A B) ! L ∼ = Rj∗ RH om(F ⊗L A B, Rj L ⊗A B).

To prove our assertion, it suffices to show that the canonical morphism L L RH om(F , M ) ⊗L A B → RH om(F ⊗A B, M ⊗A B) b is an isomorphism for any M ∈ ob Dtf (Y, A). The problem is local with respect to the etale topology. We may assume that F is a constant sheaf. When F is the constant sheaf associated to A, the above morphism is clearly an isomorphism. It follows that it is an isomorphism if F is a constant sheaf associated to a free A-module of finite rank. In general, F has a resolution by constant sheaves associated to free A-modules of finite rank. So the above morphism is an isomorphism.

Proposition 10.1.18. (i) Let f : X → Y be a morphism between noetherian schemes. For any λ-adic sheaf F = (Fn ) on Y , f ∗ F = (f ∗ Fn ) is a λ-adic sheaf on X. (ii) Let f : X → Y be an S-compactifiable morphism between noetherian schemes. For any λ-adic sheaf F = (Fn ) on X, Ri f! F = (Ri f! Fn ) are A-R λ-adic sheaves on Y for all i. (iii) Let S be a noetherian regular scheme of dimension ≤ 1, X and Y two S-schemes of finite type, f : X → Y an S-compactifiable morphism, F = (Fn ) and G = (Gn ) λ-adic sheaves on X, and H = (Hn ) a λ-adic sheaf on Y . Define Ri f∗ F = (Ri f∗ Fn ), Ri f! F = (Ri f! Fn ), f ∗ H = (f ∗ Hn ), Ri f ! H = (Ri f ! Hn ), R/(λn+1 )

TorR i (F , G ) = (Tori E xtiR (F , G )

=

(Fn , Gn )), i (E xtR/(λn+1 ) (Fn , Gn )).

Then Ri f∗ F and Ri f! F are A-R λ-adic sheaves on Y , and f ∗ H , Ri f ! H , i TorR i (F , G ) and E xtR (F , G ) are A-R λ-adic sheaves on X.

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(iv) Let S be a trait, X an S-scheme of finite type, η the generic point of S, s the closed point of S, F = (Fn ) a λ-adic sheaf on Xη , and G = (Gn ) a λ-adic sheaf on X. Define Ri Ψη (F ) = (Ri Ψη (Fn )),

Ri Φ(G ) = (Ri Φ(Gn )).

Then Ri Ψη (F ) and Ri Φ(G ) are A-R λ-adic sheaves on Xs¯. Proof. We prove (ii) and leave the rest to the reader. By 10.1.9, there exists an A-R exact sequence φ

ψ

0 → F 0 → F → F 00 → 0

such that F 0 is a torsion λ-adic sheaf, and F 00 is a torsion free λ-adic sheaf. We have short exact sequences 0 0 0 0

→ ker φn → im φn → ker ψn → im ψn

→ Fn0 → ker ψn → Fn → Fn00

→ im φn → ker ψn /im φn → im ψn → Fn00 /im ψn

→ 0, → 0, → 0, → 0.

Moreover, (ker φn ), (ker ψn /im φn ), and (Fn00 /im ψn ) are null systems. Hence (Ri f! ker φn ), (Ri f! (ker ψn /im φn )), and (Ri f! (Fn00 /im ψn )) are null systems. In particular, they are A-R λ-adic. Since F 0 is a torsion λ-adic sheaf, (Ri f! Fn0 ) are also A-R λ-adic. By 10.1.7 (iii) and the long exact sequences for R· f! , we see that to prove that (Ri f! Fn ) are A-R λ-adic, it suffices to prove that (Ri f! Fn00 ) are A-R λ-adic. But (Fn00 ) ∈ ob Dcb (X, R). By 10.1.17, we have (Rf! Fn00 ) ∈ ob Dcb (Y, R). By 10.1.16, (Ri f! Fn00 ) are A-R λ-adic. Let (Dn , Tn+1 )n≥0 be a family such that Dn are triangulated categories, and Tn+1 : Dn+1 → Dn are exact functors. Define the inverse limit D of (Dn , Tn )n≥0 as follows: Objects in D are families (Kn , un )n≥0 , where Kn ∈ ∼ = ob Dn , and un are isomorphisms Tn (Kn ) → Kn−1 in Dn−1 . Given objects K = (Kn , un ) and L = (Ln , vn ) in D, a morphism f = (fn ) : K → L consists of morphisms fn : Kn → Ln in Dn such that the diagrams Tn (Kn ) un ↓ Kn−1

Tn (fn )

→

fn−1

→

T (Ln ) ↓ vn Ln−1

commute for all n. A distinguished triangle in D is a sextuple ((Kn , un ), (Ln , vn ), (Mn , wn ), (φn ), (ψn ), (δn )) such that (Kn , un ), (Ln , vn ), (Mn , wn )

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are objects in D, (φn ) : (Kn ) → (Ln ),

(ψn ) : (Ln ) → (Mn ),

(δn ) : (Mn ) → (Kn [1])

are morphisms in D, and (Kn , Ln , Mn , φn , ψn , δn ) are distinguished triangles in Dn for all n. Lemma 10.1.19. Notation as above. If for any n ≥ 0 and any K, L ∈ Dn , Hom(K, L) is finite, then D is a triangulated category. Proof. Let K = (Kn ) and L = (Ln ) be objects in D, and let φ = (φn ) : K → L be a morphism in D. For each n, by (TR1) for Dn , we have a distinguished triangle φn

Kn → Ln → Mn → in Dn , and by (TR3) for Dn−1 , we have an isomorphism of distinguished triangles T (φn )

Tn (Kn ) → Tn (Ln ) → Tn (Mn ) → ∼ ∼ ∼ =↓ = ↓ =↓ φn−1

→

Kn−1

Ln−1 → Mn−1 →

in Dn−1 . Then φ

(Kn ) → (Ln ) → (Mn ) → is a distinguished triangle in D. This proves (TR1) for D. One can easily verify (TR2) for D. Given a commutative diagram (Kn ) → (Ln ) → (Mn ) → (gn ) ↓ → (L0n ) → (Mn0 ) →

(fn ) ↓ (Kn0 )

in D, where the horizontal lines are distinguished triangles in D, let Sn (n ≥ 0) be the sets of morphisms hn : Mn → Mn0 such that (fn , gn , hn ) are morphisms of triangles in Dn : Kn → Ln → Mn → ↓ gn ↓ hn ↓ Kn0 → L0n → Mn0 → .

fn

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For each hn ∈ Sn , let πn (hn ) be the element in Sn−1 so that the following diagram commutes: Tn (Mn ) ∼ = ↓ Mn−1

Tn (hn )

→

πn (hn )

→

T (Mn0 ) ↓∼ = 0 Mn−1 .

By our assumption and (TR3) for Dn , Sn are nonempty finite sets. So limn (Sn , πn ) is nonempty. Let (hn ) be an element in limn (Sn , πn ). Then ←− ←− ((fn ), (gn ), (hn )) is a morphism of triangles. This proves (TR3) for D. Similarly, one can verify (TR4) for D. Lemma 10.1.20. Let X be a scheme of finite type over a field k. Suppose that k is either separably closed or finite. Then for any K, L ∈ b ob Dctf (X, R/(λn+1 )), Hom(K, L) is finite. Proof.

Let f : X → Spec k be the structure morphism. We have Hom(K, L) ∼ = R0 Γ(Spec k, Rf∗ RH om(K, L)).

b By 9.5.2 and 9.5.3, we have Rf∗ RH om(K, L) ∈ ob Dctf (Spec k, R/(λn+1 )). When k is separably closed, we have

R0 Γ(Spec k, Rf∗ RH om(K, L)) ∼ = Γ(Spec k, R0 f∗ RH om(K, L)) ∼ ¯ b By 4.3.7, which is finite. When k is a finite field, we have Gal(k/k) = Z. ¯ R0 Γ(Spec k, Rf∗ RH om(K, L)) ∼ Rf∗ RH om(K, L)) = H 0 (Gal(k/k),

is finite.

Corollary 10.1.21. Let X be a scheme of finite type over a field k which is either separably closed or finite. Then Dcb (X, R) is a triangulated category. Let k be a separably closed field or a finite field, X and Y two k-schemes of finite type, and f : X → Y a k-compactifiable morphism. Then Rf∗ , Rf! : Dcb (X, R) → Dcb (Y, R),

− ⊗L R −,

f ∗ , Rf ! : Dcb (Y, R) → Dcb (X, R),

RH om(−, −) : Dcb (X, R) × Dcb (X, R) → Dcb (X, R)

are exact functors. Given a distinguished triangle (Kn ) → (Ln ) → (Mn ) →

in Dcb (Y, R), we have a long A-R exact sequence

· · · → (H i (Kn )) → (H i (Ln )) → (H i (Mn )) → · · ·

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of A-R λ-adic sheaves. Using these facts, one easily gets the long exact sequences for R· f∗ and R· f! associated to distinguished triangles in Dcb (X, R). Let X be a noetherian scheme. Recall that E is the fraction field of R. In the category A-R λ-adic sheaves on X, morphisms defined by multiplication by λm (m ≥ 0) form a multiplicative system. By 6.2.1, we can construct a category so that its objects are A-R λ-adic sheaves, and for any two objects F and G , morphisms from F to G are diagrams of the form F ↓ F

λm

f

&

G

such that m ≥ 0 and f : F → G are morphisms of A-R λ-adic sheaves. We call this category the category of E-sheaves. Objects in this category are called E-sheaves. Denote the space of morphisms from F to G in this category by HomE (F , G ). Then we have HomAR (F , G ) ⊗R E ∼ = HomE (F , G ). Indeed, assign 1 ∈ HomAR (F , G ) ⊗R E λm to the morphism in HomE (F , G ) defined by the above diagram. This as1 1 signment is well-defined and surjective. If f1 ⊗ λm and f2 ⊗ λm are mapped 1 2 to the same morphism in HomE (F , G ), where f1 , f2 ∈ HomAR (F , G ) and m1 , m2 ≥ 0, then there exist f3 ∈ HomAR (F , G ), m3 ≥ 0 and A-R morphisms φ1 , φ2 : F → F such that f⊗

λm1 φ1 = λm2 φ2 = λm3 ,

f1 φ1 = f2 φ2 = f3 . We have φ1 λm1 = φ2 λm2 = λm3 . So we have f1 ⊗

1 1 = f1 λm3 ⊗ m1 +m3 λm1 λ 1 = f1 φ1 λm1 ⊗ m1 +m3 λ 1 = f3 λm1 ⊗ m1 +m3 λ 1 = f 3 ⊗ m3 . λ

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Similarly, we have f2 ⊗

1 1 = f 3 ⊗ m3 . λm2 λ

So we have 1 1 = f 2 ⊗ m2 . λm1 λ Hence the above assignment is injective. The category of E-sheaves is abelian. In this category, multiplications by λm (m ≥ 0) are isomorphisms. Since on a torsion sheaf, multiplication by some λm is 0, any torsion sheaf becomes a zero object in this category. Conversely, if an A-R λ-adic sheaf F is a zero object in the category of E-sheaves, then idF = 0 in this category. This means that multiplication on F by some λm is 0. So F is a torsion sheaf. By 10.1.9, any E-sheaf is isomorphic to a torsion free λ-adic sheaf in the category of E-sheaves. An E-sheaf is called lisse if it is isomorphic to a lisse λ-adic sheaf in the category of E-sheaves. If k is a separably closed field, then the A-R category of λ-adic sheaves on Spec k is equivalent to the category of finitely generated R-modules. For any finitely generated R-modules M and N , we have f1 ⊗

HomR (M, N ) ⊗R E ∼ = HomE (M ⊗R E, N ⊗R E). It follows that the functor M 7→ M ⊗R E defines an equivalence between the category of E-sheaves on Spec k and the category of finite dimensional E-vector spaces. Let X be a noetherian scheme, F an E-sheaf on X, and s → X a geometric point on X. Choose a λ-adic sheaf F 0 representing F . We define the stalk of F at s to be Fs = Fs0 ⊗R E. A sequence F →G →H in the category of E-sheaves is exact if and only if Fx¯ → Gx¯ → Hx¯ is exact for any x ∈ X. Let F = (Fn ) be a λ-adic sheaf on X. Consider the functor from the category of λ-adic sheaves to the category of E-sheaves that maps each λadic sheaf G = (Gn ) to F ⊗ G = (Fn ⊗R/(λn+1 ) Gn ) regarded as an E-sheaf. This functor transforms multiplications by λm on G to isomorphisms in the category of E-sheaves. So it induces a functor on the category of E-sheaves. Applying the same argument to the first variable, we get the tensor product

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functor on the category of E-sheaves which maps E-sheaves represented by λ-adic sheaves F = (Fn ) and G = (Gn ) to the E-sheaf represented by the λ-adic sheaf F ⊗ G = (Fn ⊗R/(λn+1 ) Gn ). Let F = (Fn ) be a λ-adic sheaf such that Fn are locally free sheaves of R/(λn+1 )-modules of finite rank. Consider the functor from the category of λ-adic sheaves to the category of E-sheaves that maps each λ-adic sheaf G = (Gn ) to H om(F , G ) = (H om(Fn , Gn )) regarded as an E-sheaf. This functor transforms multiplications by λm on G to isomorphisms in the category of E-sheaves. So it induces a functor on the category of Esheaves. If F 0 = (Fn0 ) is another λ-adic sheaf such that Fn0 are locally free sheaves of R/(λn+1 )-modules of finite ranks, and if F 0 → F is an isomorphism in the category of E-sheaves, then we have an isomorphism H om(F , G ) → H om(F 0 , G ) in the category of E-sheaves. On the category Dcb (X, R), morphisms defined by multiplications by m λ (m ≥ 0) form a multiplicative system. By 6.2.1, we can construct a category Dcb (X, E) so that its objects are objects in Dcb (X, R), and for any two objects K and L, morphisms from K to L are diagrams of the form K ↓ K

λm

f

&

L

such that m ≥ 0 and f : K → L are morphisms in Dcb (X, R). For any K, L ∈ ob Dcb (X, E), we have HomDcb (X,R) (K, L) ⊗R E ∼ = HomDcb (X,E) (K, L).

Let S be a noetherian regular scheme of dimension ≤ 1, X and Y two S-schemes of finite type, and f : X → Y an S-compactifiable morphism. Then we have functors Rf∗ , Rf! : Dcb (X, E) → Dcb (Y, E), − ⊗L E −,

f ∗ , Rf ! : Dcb (Y, E) → Dcb (X, E),

RH om(−, −) : Dcb (X, E) × Dcb (X, E) → Dcb (X, E).

If S = Spec k for a separably closed field or a finite field k, then Dcb (X, E) and Dcb (Y, E) are triangulated categories, and the above functors are exact. If X is a scheme of finite type over a trait S with generic point η and closed point s, then we have functors RΨη : Dcb (Xη , E) → Dcb (Xs¯, E),

RΦ : Dcb (X, E) → Dcb (Xs¯, E).

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Let E 0 be a finite extension of E, let R0 be the integral closure of R in E , and let λ0 be a uniformizer of R0 . For any λ-adic sheaf F = (Fn ) on a noetherian scheme X, define an inverse system F ⊗R R0 = (Fn0 ) as follows: Let e be the ramification index of R0 over R. For any pair n, i ≥ 0 such that 0

ie < n + 1 ≤ (i + 1)e, we have λ0n+1 R0 ∩ R = λi+1 R. We define Fn0 = Fi ⊗R/(λi+1 ) R0 /(λ0n+1 ). Then F ⊗R R0 is a λ0 -adic sheaf. We thus get a functor F 7→ F ⊗R R0

from the category of λ-adic sheaves to the category of λ0 -adic sheaves. This functor transforms multiplications by λm (m ≥ 0) to composites of multiplications by λ0me with isomorphisms in the category of λ0 -adic sheaves. So it induces a functor F 7→ F ⊗E E 0 from the category of E-sheaves to the category of E 0 -sheaves. 0 0 Let K = (Kn ) be an object in Dcb (X, R). Define K ⊗L R R = (Kn ) by 0 0n+1 Kn0 = Ki ⊗L ) R/(λi+1 ) R /(λ

for any pair n, i ≥ 0 such that ie < n + 1 ≤ (i + 1)e. Then K

⊗L R

0

R is an object in Dcb (X, R0 ). We thus get a functor Dcb (X, R) → Dcb (X, R0 ),

0 K 7→ K ⊗L RR .

It induces a functor Dcb (X, E) → Dcb (X, E 0 ),

0 K 7→ K ⊗L E E .

Fix an algebraic closure of Q` of Q` . The direct limit of the categories of E-sheaves on a noetherian scheme X as E goes over the family of finite extensions of Q` in Q` is called the category of Q` -sheaves. Objects in this category are called Q` -sheaves. Each Q` -sheaf is represented by an E-sheaf for some finite extension E of Q` in Q` . Given two Q` -sheaves represented

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by an E1 -sheaf F1 and an E2 -sheaf F2 , the space of morphisms from F1 to F2 in the category of Q` -sheaves is HomQ` (F1 , F2 ) = lim Hom(F1 ⊗E1 E, F2 ⊗E2 E), −→ E

where the direct limit is taken over those finite extensions E of Q` in Q` containing both E1 and E2 . A Q` -sheaf is called lisse if it is represented by a lisse E-sheaf for some finite extension E of Q` in Q` . Let F be a Q` -sheaf on X and let s be a geometric point. Choose an E-sheaf F 0 representing F . We define the stalk of F at s to be Fs = Fs0 ⊗E Q` . The direct limit of the categories Dcb (X, E) as E goes over the family of finite extensions of Q` in Q` is denoted by Dcb (X, Q` ). Each object in this category is represented by an object in Dcb (X, E) for some finite extension E of Q` in Q` . Given two objects in this category, represented by K1 ∈ ob Dcb (X, E1 ) and K2 ∈ ob Dcb (X, E2 ), we have L HomDb (X,Q` ) (K1 , K2 ) = lim Hom(K1 ⊗L E1 E, K2 ⊗E2 E), c −→ E

where the direct limit is taken over those finite extensions E of Q` in Q` containing both E1 and E2 . Let S be a noetherian regular scheme of dimension ≤ 1, X and Y two S-schemes of finite type, and f : X → Y an S-compactifiable morphism. Then we have functors Rf∗ , Rf! : Dcb (X, Q` ) → Dcb (Y, Q` ), L −, − ⊗Q `

f ∗ , Rf ! : Dcb (Y, Q` ) → Dcb (X, Q` ),

RH om(−, −) : Dcb (X, Q` ) × Dcb (X, Q` ) → Dcb (X, Q` ).

If S = Spec k for a separably closed field or a finite field k, then Dcb (X, Q` ) and Dcb (Y, Q` ) are triangulated categories, and the above functors are exact. If X is a scheme of finite type over a trait S with generic point η and closed point s, then we have functors RΨη : Dcb (Xη , Q` ) → Dcb (Xs¯, Q` ),

RΦ : Dcb (X, Q` ) → Dcb (Xs¯, Q` ).

Let M be a finitely generated R-module provided with the λ-adic topology. Any R-endomorphism φ : M → M is continuous. We have M∼ M/λn+1 M . So we have = lim ←−n AutR (M ) ∼ AutR/(λn+1 ) (M/λn+1 M ). = lim ←− n

Put the discrete topology on AutR/(λn+1 ) (M/λn+1 M ), and put the inverse limit topology on AutR (M ). In the case where M = Rk , there is another way to define a topology on AutR (Rk ). Representing each element in

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AutR (Rk ) by the matrix with respect to the standard basis of Rk , we can 2 embed AutR (Rk ) into Rk . Put on AutR (Rk ) the topology induced from 2 the product topology on Rk . These two topologies on AutR (Rk ) are the same since the family {A|A ∈ AutR (Rk ), A ≡ I mod λn+1 }

(n ∈ N)

are base of neighborhoods at the identity for both topologies. Lemma 10.1.22. (i) Let M be a finitely generated R-module and let G be a topological group acting linearly on M . Then the homomorphism G → AutR (M ) is continuous if and only if the action G × M → M is continuous. (ii) Let G be a topological group acting linearly on E k . Then the homomorphism G → GL(E k ) is continuous if and only if the action G×E k → E k is continuous. Proof. (i) The canonical maps AutR/(λn+1 ) (M/λn+1 M ) × M/λn+1 M → M/λn+1 M are continuous with respect to the discrete topology for all n. It follows that lim AutR/(λn+1 ) (M/λn+1 M ) × lim M/λn+1 M → lim M/λn+1 M ← − ← − ← − n n n is continuous, that is, AutR (M ) × M → M is continuous. If G → AutR (M ) is continuous, then the action G × M → M is continuous. We have commutative diagrams G×M → M ↓ ↓ n+1 n+1 G × M/λ M → M/λ M. The vertical arrows are surjective open maps. If G×M → M is continuous, then G × M/λn+1 M → M/λn+1 M are continuous. Since M/λn+1 M are finite and discrete, the homomorphisms G → AutR/(λn+1 ) (M/λn+1 M ) are continuous. So G → AutR (M ) is continuous. (ii) The canonical map GL(E k ) × E k → E k is continuous. If G → GL(E k ) is continuous, then G × E k → E k is continuous. Let {e1 , . . . , en } be the standard basis of E k , let g ∈ G, let (aij ) ∈ GL(E k ) be the matrix of g with respect to the standard basis, and let Uij

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be a neighborhood of aij in E for each pair (i, j). If G × E k → E k is continuous. then the set Gi = {h ∈ G|hei ∈ U1i × · · · × Uki }

is open in G and contains g. The map G → GL(E k ) maps the open Q neighborhood G1 ∩ · · · ∩ Gk of g in G to the set GL(E k ) ∩ (i,j) Uij . So G → GL(E k ) is continuous. Proposition 10.1.23. Let X be a connected noetherian scheme and let x be a point in X. The functor F 7→ Fx¯ defines an equivalence between the category of lisse λ-adic sheaves (resp. lisse E-sheaves) on X and the category of finitely generated R-modules (resp. finite dimensional E-vector spaces) with continuous π1 (X, x ¯)-actions. Proof. Let F = (Fn ) be a lisse λ-adic sheaf. Each Fn gives rise to a continuous homomorphism π1 (X, x ¯) → AutR/(λn+1 ) (Fn,¯x ). n+1 ∼ We have Fn,¯x = Fx¯ /λ Fx¯ . So we have a continuous homomorphism ∼ AutR (Fx¯ ). π1 (X, x¯) → lim AutR/(λn+1 ) (Fx¯ /λn+1 Fx¯ ) = ← − n Conversely, if M is a finitely generated R-module and π1 (X, x ¯) → AutR (M ) ∼ AutR/(λn+1 ) (M/λn+1 M ) = lim ← − n is a continuous homomorphism, then the continuous representations π1 (X, x ¯) → AutR/(λn+1 ) (M/λn+1 M )

define locally constant constructible sheaves of R/(λn+1 )-modules Fn on X, and F = (Fn ) is a λ-adic sheaf with Fx¯ ∼ = M . This proves that the category of lisse λ-adic sheaves is equivalent to the category of finitely generated R-modules with continuous π1 (X, x ¯)-actions. Let RepR (resp. RepE ) be the category of finitely generated R-modules (resp. finite dimensional E-vector spaces) with continuous π1 (X, x ¯)-actions, and let S be the multiplicative system in RepR consisting of multiplications by λm (m ≥ 0) on objects in RepR . Then the category S −1 RepR is equivalent to the category of lisse E-sheaves on X. Note that if M is a finitely generated R-module with a continuous π1 (X, x ¯)-action, then M ⊗R E is a finite dimensional E-vector space with a continuous π1 (X, x ¯)-action. Indeed, let Mt = {z ∈ M |λn z = 0 for some n ≥ 0}

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be the torsion submodule of M . Then Mt is stable under the action of π1 (X, x ¯). On the other hand, the canonical homomorphism AutR (M ) → AutR (M/Mt ) is continuous. It follows that M/Mt is a free R-module of finite rank with a continuous π1 (X, x ¯)-action. We have M ⊗R E ∼ = M/Mt ⊗R E. Choose a basis for M/Mt . It induces a basis for M ⊗R E. Let k be the rank of M/Mt . The composite of the canonical homomorphism GL(Rk ) → GL(E k ) and the homomorphism π1 (X, x¯) → GL(Rk ) which is defined by the action of π1 (X, x ¯) on M/Mt using the chosen basis of M/Mt , coincides with the homomorphism π1 (X, x ¯) → GL(E k ) defined by the action of π1 (X, x ¯) on M ⊗R E using the corresponding basis on M ⊗R E. It follows that π1 (X, x ¯) → GL(E k ) is continuous and hence the action of π1 (X, x ¯) on M ⊗R E is continuous. We thus get a functor RepR → RepE , m

M 7→ M ⊗R E.

It transforms multiplications by λ (m ≥ 0) to isomorphisms in RepE . So it induces a functor S −1 RepR → RepE . To prove that the category of lisse E-sheaves is equivalent to the category of finite dimensional E-vector spaces with continuous π1 (X, x ¯)-actions, it suffices to show that the functor S −1 RepR → RepE is an equivalence of categories. Note that any finitely generated R-module M with a continuous π1 (X, x ¯)-action is isomorphic to M/Mt in S −1 RepR . Indeed, there exists an integer m ≥ 0 such that λm Mt = 0. The canonical projection M → M/Mt and the homomorphism M/Mt → M induced by λm : M → M define an isomorphism M ∼ = M/Mt in S −1 RepR . So to prove that the func−1 tor S RepR → RepE is fully faithful, it suffices to show that for any free R-modules M and N of finite ranks with continuous π1 (X, x ¯)-actions, we have HomRepR (M, N ) ⊗R E ∼ = HomRepE (M ⊗R E, N ⊗R E). Note that the canonical homomorphisms M → M ⊗R E and N → N ⊗R E are injective. Regard M and N as subspaces of M ⊗R E and N ⊗R E respectively through these monomorphisms. For any morphism φ : M ⊗R E → N ⊗R E

in RepE , there exists an integer m ≥ 0 such that λm φ(M ) ⊂ N . Then φ is the image of λm φ ⊗ λ1m ∈ HomRepR (M, N ) ⊗R E. If a morphism

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ψ : M → N in HomRepR (M, N ) has the property ψ ⊗ idE = 0, then ψ = 0. This proves that the functor is fully faithful. Let V be a finite dimensional E-vector space with a continuous π1 (X, x ¯)action. Fix a basis {e1 , . . . , ek } of V and let L = Re1 + · · · + Rek . If we identify GL(V ) with GL(E k ) through this basis, then the subgroup {g ∈ GL(V )|gL = L} of GL(V ) is identified with GL(Rk ) and hence is an open subgroup. So the subgroup H = {g ∈ π1 (X, x ¯)|gL = L} is open in π1 (X, x ¯), and hence the set π1 (X, x ¯)/H is finite. Let g1 , . . . , gm ∈ π1 (X, x¯) so that π1 (X, x ¯)/H = {g1 H, . . . , gm H}, and let M = g1 L + · · · + gm L. For any g ∈ π1 (X, x ¯) and any i ∈ {1, . . . , m}, there exist j ∈ {1, . . . , m} and h ∈ H such that ggi = gj h. We then have ggi L = gj L. Hence M is stable under the action of π1 (X, x ¯). It is a free Rmodule of finite rank with a continuous π1 (X, x ¯)-action and M ⊗R E ∼ =V. −1 So the functor S RepR → RepE is essentially surjective. A Q` -representation for π1 (X, x ¯) is a homomorphism π1 (X, x ¯) → GL(V ) for some finite dimensional Q` -vector space V such that we can find a finite extension E of Q` in Q` and a finite dimensional E-vector space VE with a continuous π1 (X, x ¯)-action with the property that V ∼ = V ⊗E Q` and that the homomorphism π1 (X, x ¯) → GL(V ) is the composite π1 (X, x ¯) → GL(VE ) → GL(VE ⊗E Q` ) ∼ = GL(V ).

Corollary 10.1.24. Let X be a noetherian connected scheme and let x be a point in X. The functor F → Fx¯ defines an equivalence between the category of lisse Q` -sheaves and the category of Q` -representations of π1 (X, x ¯). 10.2

Grothendieck Ogg Shafarevich Formula

([SGA 5] VIII, X, [SGA 4 21 ] Rapport 4.1–4.4.) Suppose that Λ is a ring with the identity element 1 but not necessarily commutative. Let Λ\ be the quotient of the additive group (Λ, +) by the subgroup generated by elements of the form ab − ba (a, b ∈ Λ). For any endomorphism f : Λn → Λn on a free (left) Λ-module Λn , if f is represented by the matrix (aij ) with respect to the standard basis of Λn , we define the n P trace Tr(f ) ∈ Λ\ of f to be the image of aii in Λ\ . If f : Λn → Λm and i=1

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g : Λm → Λn are two homomorphisms between free Λ-modules, we have Tr(f g) = Tr(gf ). Let f : P → P be an endomorphism on a finitely generated projective Λ-module P . We can find an integer n ≥ 0 and homomorphisms a : P → Λn and b : Λn → P such that ba = id. We define Tr(f ) to be Tr(af b). This definition does not depend on the choices of a and b. Indeed, if c : P → Λm and d : Λm → P are homomorphisms such that dc = id. Then we have Tr(af b) = Tr(adcf b) = Tr(cf bad) = Tr(cf d). Let P and Q be two finitely generated projective Λ-modules and let f : P → Q and g : Q → P be two homomorphisms. Then we have Tr(f g) = Tr(gf ). Let P · be a bounded complex of finitely generated projective Λ-modules, and let f = (f i ) : P · → P · be a morphism of complexes. We define X Tr(f ) = (−1)i Tr(f i ). i

If f is homotopic to 0, then Tr(f ) = 0. Indeed, if hi : P i → P i−1 are homomorphisms such that hi+1 di + di−1 hi = f i , then we have X Tr(f ) = (−1)i Tr(f i ) i

=

X

(−1)i Tr(hi+1 di + di−1 hi )

i

=

X

(−1)i Tr(hi+1 di ) +

i

=

X i

= 0.

X

(−1)i Tr(di−1 hi )

i

i

i i+1

(−1) Tr(d h

)+

X

(−1)i Tr(di−1 hi )

i

Any bounded complex of finitely generated projective Λ-modules is called perfect. Let Kperf (Λ) be the category whose objects are perfect complexes and whose morphisms are homotopy classes of morphisms between complexes. By the dual version of 6.2.8, the functor Kperf (Λ) → Db (Λ) is fully faithful, where Db (Λ) is the derived category of bounded complexes b of Λ-modules. Let Dperf (Λ) be the essential image of this functor. By the b above discussion, for any endomorphism f : K → K in Dperf (Λ), we can define Tr(f ). Proposition 10.2.1. Let Λ be a left noetherian ring with the identity eleb ment and let K be an object in D− (Λ). Then K ∈ ob Dperf (Λ) if and only if H i (K) are finitely generated Λ-modules and K has finite Tor-dimension.

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Proof.

Use the same argument as 6.4.6 and 10.2.2 below.

Lemma 10.2.2. Let Λ be a ring with the identity element, and let P be a flat Λ-module with finite presentation. Then P is projective. Proof. Let M → N be an epimorphism of Λ-modules. We need to show the map HomΛ (P, M ) → HomΛ (P, N ) is surjective. Suppose this is not true. Embed the cokernel of this homomorphism into an injective Z-module I. Then the homomorphism HomZ (HomΛ (P, N ), I) → HomZ (HomΛ (P, M ), I) is not injective. We have a commutative diagram HomZ (N, I) ⊗Λ P → HomZ (M, I) ⊗Λ P ↓ ↓ HomZ (HomΛ (P, N ), I) → HomZ (HomΛ (P, M ), I). The lower horizontal arrow is not injective. Since M → N is surjective and P is flat, the top horizontal arrow is injective. We will show that the vertical arrows are isomorphisms. We thus get a contradiction. Choose an exact sequence Λm → Λn → P → 0. Consider the commutative diagram HomZ (M,I)⊗Λ Λm ↓

→

HomZ (M,I)⊗Λ Λn

→

↓

HomZ (M,I)⊗Λ P

→0

↓

HomZ (HomΛ (Λm ,M),I) → HomZ (HomΛ (Λn ,M),I) → HomZ (HomΛ (P,M),I) →0.

The two horizontal lines are exact. The first two vertical arrows are isomorphisms. So HomZ (M, I) ⊗Λ P → HomZ (HomΛ (P, M ), I) is an isomorphism.

Let A be an abelian category. Define the category F (A ) of finitely filtered objects as follows: Objects in F (A ) are objects A in A provided with decreasing filtrations · · · ⊃ F i A ⊃ F i+1 A ⊃ · · ·

such that F i A = A for i 0 and F i A = 0 for i 0. Given two objects (A, F · A) and (B, F · B) in F (A ), a morphism f : (A, F · A) → (B, F · B) in

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F (A ) is a morphism f : A → B in A such that f (F i A) ⊂ F i B for all i. Then F (A ) is an additive category. Taking the i-th successive quotient defines a functor Gri : F (A ) → A ,

(A, F · ) 7→ Gri (A) = F i A/F i+1 A.

Define the category KF (A ) of finitely filtered complexes as follows: Objects in KF (A ) are complexes (K, F ) of objects in F (A ) so that there exist i0 and i1 with the property that F i K j = K j for all i ≤ i0 and all j, and F i K j = 0 for all i ≥ i1 and all j. Morphisms in KF (A ) are morphisms of complexes preserving filtrations, and two morphisms of complexes are considered to be the same morphism in KF (A ) if there exists a homotopy that preserves filtrations between these two morphisms. Then KF (A ) is a triangulated category. Let f : K → L be a morphism in KF (A ). We say that f is a quasi-isomorphism if f induces quasi-isomorphisms Gri (f ) : Gri (K) → Gri (L) for all i. Let S be the family of all quasi-isomorphisms in KF (A ). We define the derived category of finitely filtered complexes DF (A ) to be S −1 (KF (A )). Take A to be the category of Λ-modules. Let KFperf (Λ) be the full subcategory of KF (A ) consisting of those finitely filtered bounded complexes (K, F ) such that Gri (K j ) are finitely generated projective A-modules for all i and all j. Denote DF (A ) by DF (Λ). Then the canonical functor KFperf (Λ) → DF (Λ) is fully faithful. Denote its essential image by DFperf (Λ). An object (K, F ) b in DF (Λ) lies in DFperf (Λ) if and only if Gri (K) are objects in Dperf (Λ) for b all i. If (K, F ) is an object in DFperf (Λ), then K is an object in Dperf (Λ). Let f : (K, F ) → (K, F ) be an endomorphism of an object (K, F ) in DFperf (Λ), then we have X Tr(f, K) = Tr(f, Gri (K)). i

Let X be a scheme and let A be a commutative noetherian ring such that nA = 0 for some integer n. Take A to be the category of sheaves of Amodules on X, and denote DF (A ) by DF (X, A). For any bounded below finitely filtered complex K of sheaves of A-modules on X, the complex C · (K) defined by the Godement resolution is a bounded below finitely filtered complex of flasque sheaves of A-modules and each Gri (C · (K)) is

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isomorphic to the Godement resolution of Gri (K). We can define the right derived functor RΓ(X, −) : D+ F (X, A) → DF (A) of the functor Γ(X, −) and we have RΓ(X, K) ∼ = Γ(X, C · (K)), Gri RΓ(X, K) ∼ = RΓ(X, Gri (K)). If X is a compactifiable scheme over an algebraically closed field, and let j : X ,→ X be a compactification of X, then we have a functor RΓc (X, −) : D+ F (X, A) → DF (A) defined by RΓc (X, −) = RΓ(X, j! −). Suppose that X is a compactifiable scheme over an algebraically closed field k. Let j : U → X an open immersion, l : X − U → X a closed b immersion, and K ∈ ob Dctf (X, A). Provide K with the following filtration  if i ≤ 0, K F i K = j! j ∗ K if i = 1,  0 if i ≥ 2.

Then K can be considered as an object in D+ F (X, A), and we have  ∗  l∗ l K if i = 0, i ∼ Gr (K) = j! j ∗ K if i = 1,  0 if i 6= 0, 1. So we have

  RΓc (X − U, K|X−U ) Gri RΓc (X, K) ∼ = RΓc (X, Gri (K)) ∼ = RΓc (U, K|U )  0

if i = 0, if i = 1, if i 6= 0, 1.

By 10.2.1, 7.4.7 (ii) and 7.8.1, RΓc (X − U, K|X−U ) and RΓc (U, K|U ) are b objects in Dperf (A). So RΓc (X, K) is an object in DFperf (A). Let f : X → X be a k-morphism such that f −1 (U ) = U , and let f ∗ : f ∗ K → K be a morphism of complexes. Then f induces a morphism f ∗ K → K in DF (X, A) and hence an endomorphism f ∗ : RΓc (X, K) → RΓc (X, K)

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in DFperf (A). We have X Tr(f ∗ , RΓc (X, K)) = Tr(f ∗ , Gri RΓc (X, K)) i

= Tr(f ∗ , RΓc (U, K|U )) + Tr(f ∗ , RΓc (X − U, K|X−U )).

Suppose that D is a triangulated category so that isomorphic classes of objects in D form a set. Let L be the free abelian group generated by isomorphic classes of objects in D. For any object X in D, let (X) be the element in L corresponding to the isomorphic class of X. Let R be the subgroup of L generated by elements of the form (X) − (Y ) + (Z) whenever there exists a distinguished triangle X →Y →Z→. Define the K-group of D to be K(D) = L/R. For any X ∈ ob D, denote by [X] the image of (X) in K(D). For any abelian group G and any map φ : {isomorphic classes of objects in D} → G with the property

φ (X) − φ (Y ) + φ (Z) = 0

for any distinguished triangle

X → Y → Z →, there exists a unique homomorphism ψ : K(D) → G such that ψ([X]) = φ (X) . If D → D 0,

D × D 0 → D 00

are exact functors for some triangulated categories D, D 0 and D 00 so that isomorphic classes of objects in these categories form sets, then these functors induce homomorphisms K(D) → K(D 0 ),

K(D) × K(D 0 ) → K(D 00 ).

Let Λ be a ring with the identity element and let L0 be the free abelian group generated by isomorphic classes of finitely generated projective Λmodules. For any finitely generated projective Λ-module M , let (M ) be the element in L0 corresponding to the isomorphic class of M . Let R0 be the subgroup of L0 generated by elements of the form (M 0 ) − (M ) + (M 00 ) whenever there exists a short exact sequence of finitely generated projective Λ-modules 0 → M 0 → M → M 00 → 0.

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Define the K-group of Λ to be K(Λ) = L0 /R0 . For any finitely generated projective Λ-module M , denote by [M ] the image of (M ) in K(Λ). Proposition 10.2.3. Let Λ be a ring with the identity element. We have b an isomorphism K(Dperf (Λ)) ∼ = K(Λ). b Proof. We know Dperf (Λ) is equivalent to the category Kperf (Λ) of perfect complexes. It suffices to show that K(Kperf (Λ)) ∼ = K(Λ). For any perfect complex

let φ(P · ) =

P i

P · = (· · · → P i → P i+1 → · · · ), (−1)i [P i ] ∈ K(Λ). We claim that φ(P · ) only depends on the

isomorphic class of P · in Kperf (Λ). Let f : P · → Q· be an isomorphism in Kperf (Λ) and let C · be the mapping cone of f . We have C i = P i ⊕ Qi+1 for all i. It follows that φ(P · ) − φ(Q· ) = φ(C · ). To prove our assertion, it suffices to show φ(C · ) = 0. Note that C · is a bounded acyclic complex of projective Λ-modules. Suppose that it is of the form 0 → C a → C a+1 → · · · → C a+m → 0. Let Z i = Ker (C i → C i+1 ). We have short exact sequences 0 → Z a+m−1 → C a+m−1 → C a+m → 0, 0 → Z a+m−2 → C a+m−2 → Z a+m−1 → 0, .. . 0 → Z a+2 0 → Ca

→ C a+2 → C a+1

→ Z a+3 → Z a+2

→ 0, → 0.

One can show that Z i are finitely generated projective Λ-modules. We have X φ(C · ) = (−1)i [C i ] i

=

X

(−1)i ([Z i ] + [Z i+1 ])

i

=

X i

= 0.

(−1)i [Z i ] +

X

(−1)i [Z i+1 ]

i

This proves our assertion. So φ defines a map on the set of isomorphic classes of objects in Kperf (Λ). Any distinguished triangle in Kperf (Λ) is isomorphic to a distinguished triangle P1· → P2· → P3· →

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such that P3· is the mapping cone of the morphism P1· → P2· . We have P3i = P1i ⊕ P2i+1 for all i. It follows that φ(P1· ) − φ(P2· ) + φ(P3· ) = 0. So φ induces a homomorphism Φ : K(Kperf (Λ)) → K(Λ).

For any finitely generated projective Λ-module P , let ψ(P ) be the element in K(Kperf (Λ)) corresponding to the isomorphic class of the complex ··· → 0 → P → 0 → ··· .

ψ(P ) depends only on the isomorphic class of P and ψ induces a homomorphism Ψ : K(Λ) → K(Kperf (Λ)).

One can easily verify ΦΨ = id. Let us prove ΨΦ = id, that is, for any perfect complex P · , we have the equality X [P · ] = (−1)i ψ(P i ) i

in K(Kperf (Λ)). We say that P · has length ≤ m if the exists an integer a such that P i = 0 for i 6∈ [a, a + m], that is, P · is of the form 0 → P a → P a+1 → · · · → P a+m → 0.

We prove the above equality using induction on m. First note that we have a distinguished triangle of the form It follows that

P · → 0 → P · [1] → . [P · [1]] = −[P · ].

Suppose m = 0. Then P · coincides with the complex (· · · → 0 → P a → 0 → · · · )[a],

where P a sits in the 0-th components. It follows that [P · ] = (−1)a ψ(P a ). Suppose that we have shown the above equality for perfect complexes of length ≤ m − 1. Let σ≥a+1 (P · ) be the truncated complex 0 → P a+1 → · · · → P a+m → 0.

We have a morphism of complex

P a [−(a + 1)] → σ≥a+1 (P · )

whose (a + 1)-th component is the morphism daP · : P a → P a+1 . The mapping cone of this morphism is isomorphic to P · , and σ≥a+1 (P · ) has length ≤ m − 1. So we have X [P · ] = −[P a (−(a + 1))] + [σ≥a+1 (P · )] = (−1)i ψ(P i ). i

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Let X be a smooth irreducible projective curve over an algebraically closed field k, let K be its function field, and let K 0 be a finite galois extension of K with galois group G. Denote by X 0 the normalization of X in K, and by p : X 0 → X the canonical morphism. Then G acts on X 0 on the right. For any closed point x0 in X 0 , let Gx0 be the stabilizer of x0 in G, let π be a uniformizer of OX 0 ,x0 , and let vx0 be the valuation on K 0 corresponding to the point x0 . The Swan character on Gx0 is defined to be   1 − vx0 (gπ − π) if g 6= e, P bx0 (g) = − (1 − vx0 (gπ − π)) if g = e.  g∈Gx0 −{e}

By [Serre (1979)] VI 2 and [Serre (1977)] 19.2 Theorem 44, there exists a projective Zl [Gx0 ]-module Swx0 determined up to isomorphism such that b is its character. The Artin character of Gx0 is defined to be ax0 = bx0 + rGx0 − 1 = bx0 + uGx0 , where rGx0 is the character of the regular representation Q` [Gx0 ] of Gx , and uGx0 is the character of the kernel of the homomorphism X X Q` [Gx0 ] → Q` , ag g 7→ ag . g∈Gx0

We have

ax0 (g) =

  

g∈Gx0

−v 0 (gπ − π) P x vx0 (gπ − π)

if g 6= e, if g = e.

g∈Gx0 −{e}

The Artin character is the character of a Q` [Gx0 ]-module. Let x = p(x0 ). Define Swx = Z` [G] ⊗Z` [Gx0 ] Swx0 . Choose g1 , . . . , gm ∈ G such that G/Gx0 = {g1 Gx0 , . . . , gm Gx0 }. The character of Swx is X bx (g) = bx0 (gi−1 ggi ). gi−1 ggi ∈Gx0

We have −1 p−1 (x) = {x0 g1−1 , . . . , x0 gm }.

Set x0i = x0 gi−1 . We have Gx0i = gi Gx0 gi−1 , and vx0i (α) = vx0 (gi−1 α) for any α ∈ K. It follows that bx0i (g) = bx0 (gi−1 ggi ) for any g ∈ Gx0i . Therefore X X bx (g) = bx0i (g) = bx0 (g). g∈Gx0

i

p(x0 )=x, gx0 =x0

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This shows that Swx = Z` [G] ⊗Z` [Gx0 ] Swx0 depends only on x, and not on the choice of x0 ∈ p−1 (x). Lemma 10.2.4. Let X be a smooth irreducible projective curve of genus g over an algebraically closed field k, K its function field, K 0 a finite galois extension of K with galois group G, X 0 the normalization of X in K 0 , p : X 0 → X the canonical morphism, U a dense open subset of X such that U 0 = p−1 (U ) is etale over U , and j 0 : U 0 → X 0 the open immersion. Then b RΓ(X 0 , j!0 Z/`n ) ∈ ob Dperf (Z/`n [G]) and RΓ(X 0 , j!0 Z/`n ) X = (2 − 2g) Z/`n [G] − Z/`n [G] ⊗Z` [G] Swx + Z/`n [G] x∈X−U

b in K(Dperf (Z/`n [G])).

Proof. We have RΓ(X 0 , j!0 Z/`n ) = RΓ(X, p∗ j!0 Z/`n ). By 10.2.1, 7.4.7 (ii) and 7.8.1 applied to RΓ(X, p∗ j!0 Z/`n ), to prove RΓ(X 0 , j!0 Z/`n ) ∈ b b ob Dperf (Z/`n [G]), it suffices to prove p∗ j!0 Z/`n ∈ ob Dctf (X, Z/`n [G]). 0 There exists an etale covering {Vi → U }i such that each U ×U Vi → Vi is a trivial etale covering space. We have (p∗ j!0 Z/`n )|Vi ∼ = Z/`n [G]. In particular, (p∗ j!0 Z/`n )|U is locally constant. Moreover, we have b (p∗ j!0 Z/`n )|X−U = 0. It follows that p∗ j!0 Z/`n ∈ ob Dctf (X, Z/`n [G]). By [Serre (1977)] 14.4, we have a canonical isomorphism ∼ =

b b K(Dperf (Z` [G])) → K(Dperf (Z/`n [G])).

Denote by S the inverse of this isomorphism. By [Serre (1977)] 14.1 and 16.1 b Corollary 2 of Theorem 34, two objects in Dperf (Z` [G]) have the same image b in K(Dperf (Z` [G])) if and only if they have the same character. Denote the left-hand and the right-hand sides of the equality in the lemma by Ln and Rn , respectively. To prove the lemma, it suffices to show that S(Ln ) and S(Rn ) have the same character. Obviously S(Rn ) has character X (2 − 2g)rG − (bx + rG ), x∈X−U

where rG is the character of the regular representation Q` [G] of G. Since n ∼ 0 n RΓ(X 0 , j! Z/`m ) ⊗L Z/`m Z/` = RΓ(X , j! Z/` )

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for all m ≥ n, the character of S(Ln ) is the map G → Z` = lim Z/`n , g 7→ Tr g, RΓ(X 0 , j! Z/`n ) . ←− n

So to prove the lemma, it suffices to show X Tr g, RΓ(X 0 , j!0 Z/`n ) ≡ (2 − 2g)rG − (bx + rG ) (g) mod `n x∈X−U

for all n ≥ 0 and all g ∈ G. By 10.2.1, and 7.4.7 (ii) and 7.8.1, the complexes RΓ(X 0 , j!0 Z/`n ), RΓ(X 0 , Z/`n ) and RΓ(X 0 −U 0 , Z/`n ) are objects b in Dperf (Z/`n ). We have Tr g, RΓ(X 0 , j!0 Z/`n ) = Tr g, RΓ(X 0 , Z/`n ) − Tr g, RΓ(X 0 − U 0 , Z/`n )

for any g ∈ G. It is clear that

Tr g, RΓ(X 0 − U 0 , Z/`n ) =

X

1.

x0 ∈X 0 −U 0 , gx0 =x0

Since U 0 is etale over U , any g 6= e has no fixed point on U 0 . So by 8.6.8, if g 6= e, we have X Tr g, RΓ(X 0 , Z/`n ) ≡ (1 − bx0 (g)) mod `n gx0 =x0

≡

X

x0 ∈X 0 −U 0 ,

So for any g 6= e, we have

Tr g, RΓ(X 0 , j!0 Z/`n ) X ≡ (1 − bx0 (g)) − x0 ∈X 0 −U 0 , gx0 =x0

≡−

X

x0 ∈X 0 −U 0 ,

gx0 =x0

(1 − bx0 (g)) mod `n .

X

1 mod `n

x0 ∈X 0 −U 0 , gx0 =x0

bx0 (g) mod `n

gx0 =x0

X ≡ (2 − 2g)rG − (bx + rG ) (g)

mod `n .

x∈X−U

0

0

Let g be the genus of X . By 7.2.9 (ii), we have Tr(e, RΓ(X 0 , Z/`n )) = rank H 0 (X 0 , Z/`n ) − rank H 1 (X 0 , Z/`n ) + rank H 2 (X 0 , Z/`n )

≡ 2 − 2g 0

mod `n .

It is clear that Tr(e, RΓ(X 0 − U 0 , Z/`n )) =

X

x0 ∈X 0 −U 0

1.

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So we have Tr(e, RΓ(X 0 , j! Z/`n )) ≡ 2 − 2g 0 −

X

1 mod `n .

x0 ∈X 0 −U 0

On the other hand, we have X (2 − 2g)rG − (bx + rG ) (e) x∈X−U

0

= (2 − 2g)[K : K] − 0

= (2 − 2g)[K : K] − = (2 − 2g)[K 0 : K] − = (2 − 2g)[K 0 : K] −

X

(bx (e) + [K 0 : K])

x∈X−U

X

X

(bx0 (e) + #Gx0 )

x∈X−U p(x0 )=x

X

X

(ax0 (e) + 1)

x∈X−U p(x0 )=x

X

x0 ∈X 0 −U 0

ax0 (e) −

X

1.

x0 ∈X 0 −U 0

By [Serre (1979)] IV §1 Proposition 4, we have X ax0 (e) = deg DK 0 /K , x0 ∈X 0 −U 0

where DK 0 /K is the different of K 0 /K. We thus have X (2−2g)rG − (bx +rG ) (e) = (2−2g)[K 0 : K]−deg DK 0 /K − x∈X−U

X

1.

x0 ∈X 0 −U 0

To prove the lemma, it remains to show 2 − 2g 0 = (2 − 2g)[K 0 : K] − deg DK 0 /K . This is the Hurwitz formula, which can be proved as follows: We have an exact sequence p∗ ΩX/k → ΩX 0 /k → ΩX 0 /X → 0. Note that p∗ ΩX/k and ΩX 0 /k are invertible OX 0 -modules. Since U 0 is etale over U , p∗ ΩX/k → ΩX 0 /k induces an isomorphism on stalks at the generic point of X 0 . Since the kernel of this morphism is torsion free as a subsheaf of p∗ ΩX/k , this morphism is injective. So we have an exact sequence 0 → p∗ ΩX/k → ΩX 0 /k → ΩX 0 /X → 0. The different DK 0 /K is an effective divisor of X 0 . Let IDK 0 /K be its ideal sheaf. By the proof of [Serre (1979)] III §7 Proposition 14, the kernel of the

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canonical morphism ΩX 0 /k → ΩX 0 /X is isomorphic to IDK 0 /K ⊗OX 0 ΩX 0 /k . It follows that p∗ ΩX/k ∼ = IDK 0 /K ⊗OX 0 ΩX 0 /k . Taking degrees on both sides, we get [K 0 : K]deg (ΩX/k ) = deg (IDK 0 /K ) + deg (ΩX 0 /k ). We have deg (IDK 0 /K ) = −deg DK 0 /K . By Riemann–Roch’s formula, we have deg (ΩX/k ) = 2g − 2 and deg (ΩX 0 /k ) = 2g 0 − 2. We thus have 2 − 2g 0 = (2 − 2g)[K 0 : K] − deg DK 0 /K .

Let X be a smooth irreducible curve over an algebraically closed field b k, let A be a noetherian Z/`n -algebra, and let K ∈ ob Dctf (X, A). We can represent K by a bounded complex of flat constructible sheaves of A-modules on X. Let K(X) be the function field of X, let K(X) be a separable closure of K(X), and let η be the generic point of X. There exists a finite galois extension K 0 of K(X) contained in K(X) such that the action of Gal(K(X)/K(X)) on Kη¯ factors through G = Gal(K 0 /K(X)). Let X 0 be the normalization of X in K 0 , let p : X 0 → X be the canonical morphism, and let x be a closed point of X. One can check that Hom·Z/`n [G] (Z/`n [G] ⊗Z` [G] Swx , Kη¯) is a perfect complex of A-modules. This complex is independent of the choice of the galois extension K 0 /K(X). Indeed, let K 00 /K(X) be a finite galois extension with galois group G0 containing K 0 , and let X 00 be the normalization of X in K 00 . Then we have SwX 0 /X,x ∼ = Z` [G] ⊗Z` [G0 ] SwX 00 /X,x . One can show this by checking that both sides have the same character by using [Serre (1979)] IV §1 Proposition 3. So we have Hom·Z/`n [G] (Z/`n [G] ⊗Z` [G] SwX 0 /X,x , Kη¯)

∼ = Hom·Z/`n [G0 ] (Z/`n [G0 ] ⊗Z` [G0 ] SwX 00 /X,x , Kη¯). b Consider the following elements in K(Dperf (A)): · n αx (K) = HomZ/`n [G] (Z/` [G] ⊗Z` [G] Swx , Kη¯) ,

x (K) = αx (K) + [Kη¯] − [Kx¯ ].

By 5.8.1 (ii), we may find a dense open subset U of X and a finite surjective etale morphism U 0 → U such that U 0 is connected and the components of

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K|U 0 are constant sheaves. Moreover, we can choose U 0 so that it is galois over U . In the above discussion, taking K 0 to be the function field of U 0 , we can identify U 0 with p−1 (U ). If x is a closed point of U , then αx (K) = 0 and Kx¯ ∼ = Kη¯, and hence x (K) = 0. In particular, x (K) is nonzero only for finitely many closed point x in X. Theorem 10.2.5. Let X be a smooth irreducible projective curve of genus g over an algebraically closed field, η its generic point, A a noetherian b b Z/`n -algebra, and K ∈ ob Dctf (X, A). Then RΓ(X, K) ∈ ob Dperf (A) and X RΓ(X, K) = (2 − 2g)[Kη¯] − x (K) x∈|X|

b in K(Dperf (A)), where |X| is the set of closed points in X.

b Proof. By 10.2.1, 7.4.7 (ii) and 7.8.1, we have RΓ(X, K) ∈ ob Dperf (A). Represent K by a bounded complex of flat constructible sheaves of Amodules. Choose a finite galois extension K 0 of the function field of X and a dense open subset U of X so that if X 0 is the normalization of X in K 0 and p : X 0 → X is the canonical morphism, then p is etale on U 0 = p−1 (U ) and the components of K|U 0 are constant sheaves. Let j : U ,→ X be the open immersion. We claim that

j! (K|U ) = RΓG (p∗ p∗ j! (K|U )), where ΓG is the functor F 7→ F G on the category of G-sheaves. Indeed, by the discussion at the end of §9.1, for any x ∈ X, we have ∼ RΓG (p∗ p∗ j! (K|U ))x¯ , RΓG (p∗ p∗ j! (K|U )) = G

x ¯ G

where Γ is the functor M → M on the category of G-modules. When x ∈ X − U , we have p∗ p∗ j! (K|U ) x¯ = 0. Hence RΓG (p∗ p∗ j! (K|U )) X−U = 0.

When x ∈ U , since p : U 0 → U is a galois etale covering, components of p∗ p∗ j! (K|U ) x¯ are induced G-modules and hence G RΓG (p∗ p∗ j! (K|U )) x¯ ∼ = p∗ p∗ j! (K|U ) x¯ ∼ = Kx¯ . Our claim follows. By the discussion in §9.1 and 6.3.3, we have RΓ(X, RΓG (−)) ∼ = R Γ(X, ΓG (−)) = R ΓG Γ(X, −) = RΓG RΓ(X, −).

We thus have

RΓ(X, j! (K|U )) ∼ = RΓ(X, RΓG (p∗ p∗ j! (K|U ))) ∼ = RΓG RΓ(X, p∗ p∗ j! (K|U )).

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Let j 0 : U 0 ,→ X 0 be the open immersion. By 7.4.7, we have ∼ RΓ(X, p∗ j 0 (K|U 0 )) RΓ(X, p∗ p∗ j! (K|U )) = !

∼ = RΓc (U 0 , K|U 0 ) 0 n ∼ = Kη¯ ⊗L Z/`n RΓc (U , Z/` ) ∼ = Kη¯ ⊗L n RΓ(X 0 , j 0 Z/`n ). Z/`

Here the action of G on So we have

Kη¯ ⊗L Z/`n

RΓ(X

0

, j!0 Z/`n )

!

is the diagonal action.

0 0 n RΓ(X, j! (K|U )) ∼ = RΓG Kη¯ ⊗L Z/`n RΓ(X , j! Z/` ) .

(10.1)

b By 10.2.4, we have RΓ(X 0 , j!0 Z/`n ) ∈ ob Dperf (Z/`n [G]). Represent 0 0 n RΓ(X , j! Z/` ) by a bounded complex P of finitely generated projective (Z/`n [G])-modules. We claim that the components of the complex Kη¯ ⊗Z/`n P are weakly injective G-modules. Indeed, since the components of P are direct factors of free (Z/`n [G])-modules, it suffices to show that M ⊗Z/`n Z/`n [G] is an induced G-module for any G-module M , where the G-module structure on M ⊗Z/`n Z/`n [G] is defined by

g(x ⊗ y) = gx ⊗ gy

n

for any x ∈ M and y ∈ Z/` [G]. One easily checks M ⊗Z/`n Z/`n [G] → M [G],

x ⊗ g → (g −1 (x))g

is an isomorphism of G-modules. On the other hand, we have an isomorphism of G-modules X ∼ Hom(Z[G], M ), M [G] = xg g 7→ (g 7→ xg−1 ). g∈G

This proves our claim. So we have 0 0 G n ∼ RΓG Kη¯ ⊗L ¯ ⊗Z/`n P ) , Z/`n RΓ(X , j! Z/` ) = (Kη

(10.2)

Let P ∨ = Hom·Z/`n (P, Z/`n ), where the action of G on the i-th component HomZ/`n (P −i , Z/`n ) of P ∨ is given by (gφ)(x) = φ(g −1 x) for any g ∈ G, φ ∈ HomZ/`n (P −i , Z/`n ) and x ∈ P −i . We have an isomorphism of complexes ∼ Hom· n (P ∨ , Kη¯). Kη¯ ⊗Z/`n P = (10.3) Z/` ¿From the isomorphisms (10.1)–(10.3), we get RΓ(X, j! (K|U )) ∼ = Hom·Z/`n [G] (P ∨ , Kη¯).

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b By 10.2.4, in K(Dperf (Z/`n [G])), we have [P ] = RΓ(X 0 , j!0 Z/`n [G])] X = (2 − 2g) Z/`n [G] − Z/`n [G] ⊗Z` [G] Swx + Z/`n [G] . x∈X−U

This implies that

X [P ∨ ] = (2 − 2g) Z/`n [G] − Z/`n [G] ⊗Z` [G] Swx + Z/`n [G] . x∈X−U

This can be seen by calculating the characters on both sides, and by noting that Swx and Sw∨ x have the same character. So we have RΓ(X, j! (K|U )) = Hom·Z/`n [G] (P ∨ , Kη¯) = (2 − 2g) Hom·Z/`n [G] (Z/`n [G], Kη¯) X − Hom·Z/`n [G] (Z/`n [G] ⊗Z` [G] Swx , Kη¯) x∈X−U

+ Hom·Z/`n [G] (Z/`n [G], Kη¯) X = (2 − 2g)[Kη¯] − Hom·Z/`n [G] (Z/`n [G] ⊗Z` [G] Swx , Kη¯) + [Kη¯] . x∈X−U

We have a distinguished triangle

RΓ(X, j! (K|U )) → RΓ(X, K) → RΓ(X − U, K|X−U ) → . Again by 10.2.1, and 7.4.7 (ii) and 7.8.1, all the complexes in this triangle b are objects in Dperf (A). It is clear that X RΓ(X − U, K|X−U ) = [Kx¯ ]. x∈X−U

So we have RΓ(X, K) = RΓ(X, j! (K|U )) + RΓ(X − U, K|X−U ) X · n = (2 − 2g)[Kη¯] − HomZ/`n [G] (Z/` [G] ⊗Z` [G] Swx , Kη¯) + [Kη¯] +

X

x∈X−U

x∈X−U

[Kx¯ ]

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= (2 − 2g)[Kη¯] − + [Kη¯] − [Kx¯ ]

x∈X−U

= (2 − 2g)[Kη¯] − = (2 − 2g)[Kη¯] −

X X

Hom·Z/`n [G] (Z/`n [G] ⊗Z` [G] Swx , Kη¯)

x (K)

x∈X−U

X

x (K).

x∈|X|

Theorem 10.2.6. Let X be a smooth irreducible projective curve of genus g over an algebraically closed field, η its generic point, A a noetherian b Z/`n -algebra, U a dense open subset of X, and K ∈ ob Dctf (U, A). Then b RΓc (U, K) and RΓ(U, K) are objects in Dperf (A), and RΓc (U, K) = RΓ(U, K) X X = (2 − 2g − #(X − U ))[Kη¯] − x (K) − αx (K). x∈X−U

x∈|U|

in

b K(Dperf (A)),

where |U | is the set of closed points in U .

Proof. Let j : U ,→ X be the open immersion. By 10.2.5, we have b RΓc (U, K) = RΓ(X, j! K) ∈ ob Dperf (A) and RΓc (U, K) = RΓ(X, j! K) X X x (j! K) − x (j! K) = (2 − 2g)[Kη¯] − x∈X−U

x∈|U|

= (2 − 2g)[Kη¯] −

X

x∈|U|

x (j! K) −

= (2 − 2g − #(X − U ))[Kη¯] −

X

αx (j! K) + [Kη¯]

x∈X−U

X

x∈|U|

x (K) −

X

αx (K).

x∈X−U

b It remains to prove RΓ(U, K) ∈ ob Dperf (A) and RΓ(X, K) . Choose a distinguished triangle

RΓc (X, K) =

j! K → Rj∗ K → C →,

∼ Rj∗ j ∗ (j! K) is the canonical morphism. We have a where j! K → Rj∗ K = distinguished triangle RΓ(X, j! K) → RΓ(X, Rj∗ K) → RΓ(X, C) →,

which can be identified with a distinguished triangle

RΓc (U, K) → RΓ(U, K) → RΓ(X, C) → .

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b To prove our assertion, it suffices to show that RΓ(X, C) ∈ ob Dperf (A) and RΓ(X, C) = 0. The cohomology sheaves of C are supported on X − U , so we have X RΓ(X, C) = [Cx¯ ]. x∈X−U

b We are reduced to show that Cx¯ ∈ ob Dperf (A) and [Cx¯ ] = 0 for any ex¯ of X at x ∈ X − U . Let η˜ be the generic point of the strict localization X x ¯. By 5.9.5, we have

ex¯ , K) = RΓ(˜ Cx¯ = RΓ(U ×X X η , K) = RΓGal(¯η /˜η) (Kη¯).

Let P be the wild inertia subgroup of Gal(¯ η /˜ η ). Then P is a pro-p-group for p = char k, and we have Gal(¯ η /˜ η )/P ∼ =

lim Z/m, ←−

(m,p)=1

RΓGal(¯η/˜η) (Kη¯ ) ∼ = RΓGal(¯η/˜η)/P (Kη¯P ). b We have Kη¯P ∈ ob Dperf (A) since Kη¯P is a direct factor of Kη¯. Indeed, for any A-module M with continuous P -action, the inclusion M P ,→ M has a left inverse M → M P defined by X 1 gx, x 7→ #(P/Px ) gPx ∈P/Px

1 where Px is the stabilizer of x in P . Here multiplication by #(P/P makes x) sense since p is invertible in A. To prove our assertion, it suffices to show that for G = lim(m,p)=1 Z/m and for any complex L· of A[G]-modules ←− b which lies in Dperf (A) considered as a complex of A-modules, we have G · b b RΓ (L ) ∈ ob Dperf (A) and RΓG (L· ) = 0 in K(Dperf (A)). Represent L· G by a bounded below complex of RΓ -acyclic A-modules with continuous G-action, and let σ be the element in G so that its image in each Z/m is 1. We have

RΓG (L· ) = ker (σ − 1, L· ).

On the other hand, for each component Li of L· , we have coker (σ − 1, Li ) = H 1 (G, Li ) = 0.

by 4.3.8. So σ : Li → Li is onto. We thus have a distinguished triangle σ−1

RΓG (L· ) → L· → L· → . b It follows that RΓG (L· ) ∈ ob Dperf (A) and RΓG (L· ) = 0.

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Assume A is a local ring. Then every projective A-module of finite type is free and we may talk about its rank. It defines a homomorphism rank : K(A) → Z. By 10.2.3, we can define a homomorphism b rank : K(Dperf (A)) → Z.

b Representing an object in Dperf (A) by a bounded complex of projective A-modules K, we have X rank K = (−1)i rank K i . i

Corollary 10.2.7 (Grothendieck Ogg Shafarevich Formula). Let X be a smooth irreducible projective curve of genus g over an algebraically closed field, η its generic point, U an open dense subset, K ∈ ob Dcb (U, Q` ), and x any closed point in X. Define X χc (U, K) = (−1)i dim Hci (U, K), i

χ(U, K) =

X

(−1)i dim H i (U, K),

i

dim Kη¯ =

X

(−1)i dim (H i (K))η¯ ,

i

dim Kx¯ =

X

(−1)i dim (H i (K))x¯ ,

i

αx (K) =

X i

(−1)i dim HomG (Q` [G] ⊗Z` [G] Swx , H i (K)η¯ ),

x (K) = αx (K) + dim Kη¯ − dim Kx¯ . Then we have χc (U, K) = χ(U, K) = (2−2g−#(X−U ))dim Kη¯−

X

x (K)−

x∈|U|

X

αx (K).

x∈X−U

Proof. Let E be a finite extension of Q` such that K is represented by an object in Dcb (U, E), let R be the integral closure of Z` in E, and let λ be a uniformizer of R. Represent K by an object (Kn ) in Dcb (X, R), where b Kn ∈ ob Dctf (U, R/(λn+1 )) and Kn+1 ⊗L n+2 R/(λn+1 ) ∼ = Kn . R/(λ

We have RΓc (U, Kn ) ∈

)

b ob Dperf (R/(λn+1 ))

and

n+1 ∼ RΓc (U, Kn+1 ) ⊗L ) = RΓc (U, Kn ). R/(λn+2 ) R/(λ

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Representing each RΓc (U, Kn ) by a bounded complex Ln of free R/(λn+1 )modules of finite rank, we have quasi-isomorphisms Ln+1 ⊗R/(λn+2 ) R/(λn+1 ) ∼ = Ln . By 10.1.15, we may find a bounded complex L of free R-modules of finite ranks and quasi-isomorphisms L/λn+1 L → Ln such that the diagrams L/λn+2 L → Ln+1 ↓ ↓ L/λn+1 L → Ln commute up to homotopy. Then we have X χc (U, K) = (−1)i dim (Li ⊗R E) i

=

X

(−1)i rank (Li /λn+1 Li )

i

= rank RΓc (U, Kn ) for all n. Similarly, we have χ(U, K) = rank RΓ(U, Kn ), dim Kη¯ = rank Kn,¯η , αx (K) = rank αx (Kn ), x (K) = rank x (Kn ). We then apply 10.2.6 to Kn . 10.3

Frobenius Correspondences

([SGA 5] XIV=XV.) Let p be a prime number. A scheme is called to be of characteristic p if p · 1 = 0 in Γ(X, OX ), or equivalently, the canonical morphism X → Spec Z factors through a morphism X → Spec Z/p. Fix a power q of p. Then the morphism φ : OX → OX that maps each section s of OX to sq is a morphism of sheaves of rings. The pair (id, φ) : (X, OX ) → (X, OX ) is a morphism of schemes. We call it the Frobenius morphism on X, and denote it by frX . Let X be a scheme of characteristic p. Then every X-scheme is also of characteristic p. For any X-scheme Y , let Y (q/X) = Y ×X,frX X. We have

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a Cartesian diagram π

Y (q/X) →1 Y π2 ↓ ↓ fr

X X → X, where π1 and π2 are the projections. We have a commutative diagram

fr

Y Y → Y ↓ ↓

fr

X X → X, We define the Frobenius morphism FrY /X of Y relative to X to be the X-morphism FrY /X : Y → Y (q/X) such that π1 FrY /X = frY .

Proposition 10.3.1. Let X be a scheme of characteristic p and let Y be an X-scheme. The relative Frobenius morphism FrY /X : Y → Y (q/X) is integral, radiciel, and surjective. If Y is etale over X, then FrY /X is an isomorphism. Proof. It is clear that frX and frY and π1 are integral, radiciel and surjective. It follows that FrY /X has the same property. If Y is etale over X, then Y (q/X) is also etale over X and hence FrY /X is etale. By 2.3.9, FrY /X is an isomorphism. Let X be a scheme of characteristic p and let F be a sheaf on X. For any etale X-scheme U , since FrU/X is an isomorphism, the restriction map F (U (q/X) ) → F (U ) is an isomorphism. We thus have an isomorphism frX∗ F → F . Its inverse F → frX∗ F defines a morphism Fr∗F : fr∗X F → F .

The pair (frX , Fr∗F ) is called the Frobenius correspondence on (X, F ). We can also define a morphism Fr∗K : fr∗X K → K for any object K in the derived category D(X) = D(X, Z) of sheaves of abelian groups on X. Proposition 10.3.2. Let X and Y be schemes of characteristic p, and let f : Y → X be a morphism. (i) For any sheaf G on Y , the composite f∗ G → f∗ frY ∗ fr∗Y G Fr∗f∗ G

f∗ frY ∗ (Fr∗ G)

→

fr∗X f∗ G

f∗ frY ∗ G ∼ = frX∗ f∗ G

induces the morphism : → f∗ G by adjunction. (ii) For any sheaf F on X, the composite fr∗Y f ∗ F ∼ = f ∗ fr∗X F

coincides with Fr∗f ∗ F : fr∗Y f ∗ F → f ∗ F .

f ∗ (Fr∗ F)

→

f ∗F

etale

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Proof. We leave it for the reader to prove (i). Let us prove (ii). Consider the diagram frX∗ fr∗X F adj

F

%

→ adj &

frX∗ (Fr∗ F )↓

adj

→ frX∗ f∗ f ∗ fr∗X F ↓frX∗ f∗ f ∗ (Fr∗F )

adj

→ frX∗ f∗ f ∗ F (1) ok f∗ f ∗ F → f∗ frY ∗ f ∗ F adj & ↑f∗ frY ∗ (Fr∗ f∗F ) f∗ frY ∗ fr∗Y f ∗ F . frX∗ F

Using the definition of the Frobenius correspondence, one can check that (1) commutes. It is clear that the other parts of the diagram commute. It follows that the composite in the proposition induces the same morphism F → f∗ frY ∗ f ∗ F as Fr∗f ∗ F , and hence they coincide. Proposition 10.3.3. Let X be a scheme of characteristic p, and let K ∈ ob D+ (X). The composite Fr∗

H i (X, K) → H i (X, fr∗X K) →K H i (X, K) is the identity for each i, where the first homomorphism is the composite adj H i (X, K) → H i (X, RfrX∗ fr∗X K) ∼ = H i (X, fr∗X K).

Proof. Let I and J be bounded below complexes of injective sheaves on X so that we have quasi-isomorphisms K → I and fr∗X I → J. We can find a diagram Fr∗

fr∗X I →I I ↓ ↓ J → J0

such that J 0 is a bounded below complex of injective sheaves, I → J 0 is a quasi-isomorphism, and the diagram commutes up to homotopy. So we have a diagram Fr∗

Γ(X, I) → Γ(X, fr∗X I) →I Γ(X, I) ↓ ↓ Γ(X, J) → Γ(X, J 0 ) which commutes up to homotopy. The composite of the morphisms Fr∗

RΓ(X, K) → RΓ(X, fr∗X K) →K RΓ(X, K)

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can be identified with the composite Γ(X, I) → Γ(X, fr∗X I) → Γ(X, J) → Γ(X, J 0 ). It is homotopic to the composite Fr∗

Γ(X, I) → Γ(X, fr∗X I) →I Γ(X, I) → Γ(X, J 0 ). We claim that the composite Fr∗

Γ(X, I) → Γ(X, fr∗X I) →I Γ(X, I) is the identity. Our assertion then follows. To prove the claim, note that the composite I → frX∗ fr∗X I

fr∗ (Fr∗ I)

→

frX∗ I

is the inverse of the morphism frX∗ (I) → I defined by the restriction (frX∗ (I))(U ) = I(U (q/X) ) → I(U )

for any etale X-scheme U . If U = X, the restriction I(U (q/X) ) → I(U ) is the identity. So the composite Fr∗

Γ(X, I) → Γ(X, frX∗ fr∗X I) →I Γ(X, frX∗ I) is the identity. This proves the claim.

Let Fq be a finite field of characteristic p with q elements, and let F be an algebraic closure of Fq . Denote the Frobenius morphism on Spec F by frF . Let X0 be a scheme over Spec Fq and let X = X0 ⊗Fq F. Denote the F-morphism frX0 ⊗ id : X0 ⊗Fq F → X0 ⊗Fq F

by FX : X → X or simply F . If X0 = AnFq , we have X = AnF , and F : AnF → AnF corresponds to the F-algebra homomorphism F[t1 , . . . , tn ] → F[t1 , . . . , tn ],

ti 7→ tqi .

So F maps a point in AnF with coordinate (a1 , . . . , an ) to the point with coordinate (aq1 , . . . , aqn ). For any scheme Y over a field k, and any extension K of k, a K-point of Y is a k-morphism Spec K → Y . The set of K-points in Y is denoted by Y (K). We have a canonical bijection X(F) ∼ = X0 (F). For any F-point t : Spec F → X0 , since frX0 ◦ t = t ◦ frF , the morphism frX0 maps t to the F-point t ◦ frF . So t is fixed by frX0 if and only if there exists an Fq -point t0 : Spec Fq → X0 such that t is the composite t

0 Spec F → Spec Fq → X0 .

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It follows that the set of fixed points of F on X(F) can be identified with X0 (Fq ). Let F0 be a sheaf on X0 , let π : X → X0 be the projection, and let F = π ∗ F0 be the inverse image of F0 . Define ∗ FF : F ∗F → F 0

to be the morphism induced from Fr∗F0 : fr∗X0 F0 → F0 by base change, that is, the composite ∗ F ∗ F = F ∗ π ∗ F0 ∼ = π ∗ frX0 F0

π ∗ (Fr∗ F0 )

→

π ∗ F0 = F .

∗ We call the pair (F, FF ) the geometric Frobenius correspondence. Simi0 ∗ larly, for any K0 ∈ ob D(X0 ), we can define a morphism FK : F ∗K → K 0 in D(X), where K = π ∗ K0 . On the other hand, we have an isomorphism (idX0 × frF )∗ F ∼ = F defined as the composite ∗ (idX × frF ) F = (idX × frF )∗ π ∗ F0 ∼ = (π ◦ (idX × frF ))∗ F0 = π ∗ F0 = F . 0

0

0

Proposition 10.3.4. Notation as above. The composite F∗

F0 fr∗X F ∼ = F ∗ (idX0 × frF )∗ F ∼ = F ∗F → F

coincides with Fr∗F : fr∗X F → F . Proof.

We have a commutative diagram

fr∗X F k ∗ ∗ frX π F0 ok (π ◦ frX )∗ F0

∼ =

F ∗ (id × frF )∗ F k ∗ ∼ = F (id × frF )∗ π ∗ F0 ok = (π ◦ (id × frF ) ◦ F )∗ F0

∼ =

F ∗F k ∗ ∗ ∼ = F π F0 ok = (πF )∗ F0

∗ FF 0

→ π ∗ F0 = F ↑ π∗ (Fr∗F0 ) ∼ = π ∗ fr∗X0 F0 ok = (frX0 ◦ π)∗ F0 .

So the composite in the proposition coincides with the composite fr∗X F = fr∗X π ∗ F0 ∼ = π ∗ fr∗X0 F0

π ∗ (Fr∗ F0 )

→

π ∗ F0 = F .

By 10.3.2 (ii), this last composite coincides with Fr∗F : fr∗X F → F .

Corollary 10.3.5. Notation as above. For any K0 ∈ ob D+ (X0 ), the composite ∗ FK

H i (X, K) → H i (X, F ∗ K) →0 H i (X, K) and the composite H i (X, K) → H i (X, (idX0 × frF )∗ K) ∼ = H i (X, K) are inverse to each other.

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Proof. Denote the homomorphisms defined by the composites in the corollary by φ and ψ, respectively. Consider the diagram H i (X, K) → H i (X, (idX0 × frF )∗ K) ∼ = H i (X, K) ↓ ↓ ↓ H i (X, fr∗X K) ∼ = H i (X, F ∗ (idX0 × frF )∗ K) ∼ = H i (X, F ∗ K) Fr∗ ↓ FK∗ 0 K ↓ i H (X, K) = H i (X, K). By 10.3.4, the lower rectangle in the diagram commutes. It is clear that the other parts of the diagram commute. By 10.3.3, the composite of the vertical arrows on the left is the identity. It follows that φψ = id. It is clear that ψ is an isomorphism. So φ and ψ are inverse to each other. Proposition 10.3.6. Let n be a positive integer, let Fqn be the extension of Fq of degree n contained in F, let X1 = X0 ⊗Fq Fqn , and let F1 be the inverse image on X1 of a sheaf F0 on X0 . Define frX1 : X1 → X1 so that it is the identity on the underlying topological spaces, and that it maps each section of OX1 to its q n -th power. Define Fr∗F1 : fr∗X1 F1 → F1 as before using q n in place of q. If we iterate n times the correspondence (frX0 , Fr∗F0 ) and take the base change of the iteration under Fqn → F, we get the correspondence ∗ : F1∗ F → F be the morphism (frX1 , Fr∗F1 ). Let F1 = frX1 ⊗ idF and let FF 1 ∗ deduced from FrF1 by the base change Fqn → F. Then the correspondence ∗ ∗ (F1 , FF ) is the n-th iteration of the correspondence (F, FF ). 1 0 Proof. Let π1 : X1 → X0 be the projection. Denote by (frnX0 , Frn∗ F0 ) the n-th iteration of (frX0 , Fr∗F0 ). By 10.3.2 (ii), the composite fr∗X1 π1∗ F0 ∼ = π1∗ (frnX0 )∗ F0 coincides with Fr∗F1 . Our assertion follows.

π1∗ (Frn∗ F0 )

→

π1∗ F0

Let x be a closed point of X0 with [k(x) : Fq ] = n and let x ¯ ∈ X(F) be an F-point of X whose image in X0 is x. Then x¯ is a fixed point of F n = F1 . ∗ The n-th iteration of the correspondence (F, FF ) induces a homomorphism 0 n∗ Fx¯ : Fx¯ → Fx¯ . Denote the closed immersion Spec k(x) → X0 by i. Then i∗ F0 is completely determined by the galois action of Gal(k(x)/k(x)) on Fx¯ . Let fx : Fx¯ → Fx¯ be the action on Fx¯ of the Frobenius substitution n α 7→ αq in Gal(k(x)/k(x)). Proposition 10.3.7. With the above notation, the homomorphisms Fx¯n∗ : Fx¯ → Fx¯ and fx : Fx¯ → Fx¯ are inverse to each other.

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Proof. Without loss of generality, assume n = 1 and hence k(x) = Fq . Denote the image of x ¯ in X0 (F) also by x¯. Then x ¯ is a fixed point of frX0 , and Fx¯∗ is the homomorphism induced by Fr∗F0 : fr∗X0 F0 → F0 on stalks at x ¯. By 10.3.2 (ii), we have a commutative diagram fr∗Fq i∗ F0 ∼ = i∗ fr∗X0 F0 Fri∗ F0 ↓ ↓ i∗ (Fr∗F0 ) ∗ i F0 = i∗ F0 . We are thus reduced to the case where X0 = Spec Fq . One checks Fx∗ = fx−1 directly in this case, or use 10.3.4. (Even though frFq is the identity, Fri∗ F0 may be nontrivial.) Remark 10.3.8. Note that 10.3.5 also follows from 10.3.7 applied to the sheaf F0 = Ri f0∗ K0 on Spec Fq , where f0 : X0 → Spec Fq is the structure morphism. Let x ¯ be the geometric point Spec F → Spec Fq , and let f = f0 ⊗idF . Using 10.3.2 (i) and taking base change, we find that the composite F∗

K Rf∗ K → Rf∗ RF∗ F ∗ K →0 Rf∗ RF∗ K ∼ = RF∗ Rf∗ K

induces the morphism ∗ FRf : F ∗ Rf∗ K → Rf∗ K 0∗ K0

by adjunction. But frFq is the identity morphism. So the above composite can be written as F∗

K Rf∗ K → Rf∗ RF∗ F ∗ K →0 Rf∗ RF∗ K ∼ = Rf∗ K.

It follows that the composite ∗ FK

H i (X, K) → H i (X, F ∗ K) →0 H i (X, K) can be identified with the homomorphism Fx¯ : (Ri f∗ K)x¯ → (Ri f∗ K)x¯ . The composite H i (X, K) → H i (X, (idX0 × frF )∗ K) ∼ = H i (X, K) can be identified with the action of the Frobenius substitution on (Ri f0∗ F0 )x¯ . So 10.3.7 implies 10.3.5.

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10.4

Etale Cohomology Theory

Lefschetz Trace Formula

([SGA 4 21 ] Rapport 4–6.) Let A be a commutative noetherian ring, let G be a finite group, and let A[G]\ be the quotient of the additive group (A[G], +) by the subgroup generated by elements of the form xy − yx (x, y ∈ A[G]). Then A[G]\ has an A-module structure, and it is the quotient of the A-module A[G] by the submodule generated by elements of the form h−1 gh−g (g, h ∈ G). For any g ∈ G, let [g] be the subset of G consisting of those elements conjugate to g. Elements in [g] have the same image g¯ in A[G]\ . Choose g1 , . . . , gk ∈ G so that G is the disjoint union of the conjugacy classes [gj ] (j = 1, . . . , k). Then A[G]\ is the free A-module with basis g¯j (j = 1, . . . , k). For any g ∈ G, the map X X A[G] → A, ah h 7→ ah h∈G

h∈[g]

is an A-module homomorphism, and it vanishes on the submodule generated by h0−1 hh0 − h (h, h0 ∈ G). So it induces a homomorphism θg : A[G]\ → A. Proposition 10.4.1. Let P be a finitely generated projective A[G]-module and let v : P → P be an A[G]-module endomorphism. For any g ∈ G, let TrA (g −1 v, P ) (resp, TrA[G] (v, P )) be the trace of g −1 v (resp. v) considered as an A-module endomorphism (resp. A[G]-module endomorphism). Choose gj ∈ G (j = 1, . . . , k) so that G is the disjoint union of the conjugacy classes [gj ]. Denote the images of gj in A[G]\ by g¯j . Then we have TrA[G] (v, P ) =

k X

θgj (TrA[G] (v, P ))¯ gj ,

j=1

TrA (g −1 v, P ) = #C(g)θg (TrA[G] (v, P )), where C(g) = {h ∈ G|gh = hg} is the center of g. Proof. Let φ : P → A[G]m and ψ : A[G]m → P be A[G]-module homomorphisms such that ψφ = id. We have TrA (g −1 v, P ) = TrA (φg −1 vψ, A[G]m ) = TrA (g −1 φvψ, A[G]m ), TrA[G] (v, P ) = TrA[G] (φvψ, A[G]m ).

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To prove the proposition, we may assume P = A[G]m . For each i ∈ {1, . . . , m}, Let ei be the element in A[G]m whose i-th component is 1 and whose other components are 0. Write m X X 0 v(ei ) = ahij h0 ej h0 ∈G

j=1

Pm P with ahij ∈ A. Then TrA[G] (v, P ) is the image of i=1 h∈G ahii h in A[G]\ . Elements in [gj ] have the same image in A[G]\ . So we have 0

TrA[G] (v, P ) =

m X k X X

ahii g¯j =

j=1

i=1 j=1 h∈[gj ]

θg (TrA[G] (v, P )) =

m X X

k X m X X

i=1 h∈[gj ]

ahii g¯j ,

ahii .

i=1 h∈[g]

Hence TrA[G] (v, P ) =

k X

θgj (TrA[G] (v, P ))¯ gj .

j=1

(Actually we have x = hand, we have

Pk

gj j=1 θgj (x)¯

(g −1 v)(hei ) = g −1 hv(ei ) =

m X X

for any x ∈ A[G]\ .) On the other 0

ahij g −1 hh0 ej =

j=1 h0 ∈G

So we have

TrA (g −1 v, P ) =

m X X

−1

ahij

gh0 0

h ej .

j=1 h0 ∈G

m X X

−1

ahii

gh

i=1 h∈G

= #C(g)

m X X

ahii

i=1 h∈[g]

= #C(g)θg (TrA[G] (v, P )).

Proposition 10.4.2. Let Λ be a commutative ring, A a commutative Λalgebra, P a finitely generated projective Λ[G]-module, and M a finitely generated A[G]-module which is projective as an A-module. Then the A[G]module M ⊗Λ P is a finitely generated projective A[G]-module, where the G-action on M ⊗Λ P is given by g(x⊗y) = gx⊗gy for any x ∈ M and y ∈ P . Let v : P → P be a Λ[G]-module endomorphism. If TrΛ[G] (v, P ) = 0, then TrA[G] (idM ⊗ v, M ⊗Λ P ) = 0.

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Proof. We may reduce to the case where P = Λ[G]m . Let (M ⊗Λ P )0 be the A[G]-module M ⊗Λ P so that the G-action is given by g(x ⊗ y) = x ⊗ gy for any x ∈ M and y ∈ P . It is clear that (M ⊗Λ P )0 is a finitely generated projective A[G]-module. One can check that φ : M ⊗Λ Λ[G]m → (M ⊗Λ Λ[G]m )0 ,

x ⊗ gei 7→ g −1 x ⊗ gei

is an A[G]-module isomorphism, where ei (i = 1, . . . , m) is the element in Λ[G]m whose i-th component is 1 and whose other components are 0. So M ⊗Λ P is a finitely generated projective A[G]-module. Write m X X v(ei ) = agij gej j=1 g∈G

with agij ∈ Λ. We have

(φ ◦ (idM ⊗ v) ◦ φ−1 )(x ⊗ ei ) = φ((idM ⊗ v)(x ⊗ ei )) m X X agij gej = φ x⊗ j=1 g∈G

=

m X

X

j=1 g∈G

agij g −1 x ⊗ gej .

m P P

Hence TrA[G] (idM ⊗ v, M ⊗Λ P ) is the image of

i=1 g∈G

agii TrA (g −1 , M )g in

A[G]\ . Let g1 , . . . , gk ∈ G such that G is the disjoint union of the conjugacy classes [g1 ], . . . , [gn ], and let g¯j be the image of gj in A[G]\ . Then we have TrA[G] (idM ⊗ v, M ⊗Λ P ) = =

k X m X X

j=1 i=1 g∈[gj ]

k X m X X j=1

On the other hand, we have 0 = TrΛ[G] (v, P ) =

agii TrA (g −1 , M )¯ gj

m X X

i=1 g∈[gj ]

agii g¯ =

i=1 g∈G \

agii TrA (gj−1 , M )¯ gj .

k X m X X j=1

i=1 g∈[gj ]

agii g¯j .

Since {¯ g1 , . . . , g¯k } is a basis of Λ[G] , this implies that m X X agii = 0. i=1 g∈[gj ]

So we have TrA[G] (idM ⊗ v, M ⊗Λ P ) = 0.

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The kernel of the homomorphism X X : A[G] → A, ag g 7→ ag g∈G

g∈G

is generated by g − e (g ∈ G) as an A-module. If we put the trivial G-action on A, then is a homomorphism of A[G]-modules. For any A[G]-module P , we have A ⊗A[G] P ∼ = PG , where PG is the space of G-coinvariants, that is, the quotient of G by the A-submodule generated by gx − x (g ∈ G, x ∈ P ). If P is a finitely generated projective A[G]-module, then PG is a finitely generated projective A-module. The homomorphism induces a homomorphism δ : A[G]\ → A. Proposition 10.4.3. Let P be a finitely generated projective A[G]-module, let u : P → P be an A[G]-module endomorphism, and let u0 : PG → PG be the endomorphism induced by u. Then TrA (u0 , PG ) = δ(TrA[G] (u, P )). Proof. We may reduce to the case where P = A[G]m . For each i ∈ {1, . . . , m}, let ei (resp. e0i ) be the element in A[G]m (resp. Am ) whose i-th component is 1 and whose other components are 0. Write u(ei ) =

m X X j=1

g∈G

with agij ∈ A. We have δ(TrA[G] (u, P )) =

agij g ej

XX i

agii .

g∈G

Since PG ∼ = A ⊗A[G] P ∼ = Am , we have X X g u0 (e0i ) = aij e0j . j

g∈G

So TrA (u0 , PG ) =

XX i

g∈G

agii = δ(TrA[G] (u, P )).

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Etale Cohomology Theory

Let X0 be a compactifiable scheme over a finite field Fq of characteristic p with q elements, A a noetherian Z/`n -algebra with (`, p) = 1, K0 ∈ b ob Dctf (X0 , A), F an algebraic closure of Fq , X = X0 ⊗Fq F, K the inverse ∗ image of K0 on X, and F : X → X and FK : F ∗ K → K the morphisms 0 ∗ ∗ induced by frX0 : X0 → X0 and FrK0 : frX0 K0 → K0 respectively by base change. Choose a compactification X 0 of X0 , let X = X 0 ⊗Fq F, and let j : X ,→ X be the open immersion. Denote by F ∗ : RΓc (X, K) → RΓc (X, K) the composite RΓc (X, K) ∼ = RΓ(X, j! K)

→ RΓ(X, F ∗ j! K) ∼ = RΓ(X, j! F ∗ K)

∗ FK 0

→ RΓ(X, j! K) ∼ = RΓc (X, K).

This morphism is independent of the choice of the compactification. By b 10.2.1, 7.4.7 and 7.8.1, we have RΓc (X, K) ∈ ob Dperf (A), and for any b geometric point x ¯ of X0 , we have Kx¯ ∈ ob Dperf (A). If x ¯ is an F-point of ∗ ∗ X fixed by F , then FK : F K → K induces a morphism Fx¯∗ : Kx¯ → Kx¯ . 0 The main result of this section is the following: Theorem 10.4.4 (Lefschetz trace formula). Notation as above. Let XF ∼ = X0 (Fq ) be the set of fixed points of F on X(F) ∼ = X0 (F). We have X Tr(Fx¯∗ , Kx¯ ) = Tr(F ∗ , RΓc (X, K)). x ¯∈X F

We first prove the following corollary of this theorem. Theorem 10.4.5. Let X0 be a compactifiable scheme over Spec Fq and let K ∈ ob Dcb (X0 , Q` ). We have X Tr(Fx¯∗ , Kx¯ ) = Tr(F ∗ , RΓc (X, K)). x ¯∈X F

Proof. Let E be a finite extension of Q` such that K is represented by an object in Dcb (X, E), let R be the integral closure of Z` in E, and let λ be a uniformizer of R. Represent K by an object (Kn ) in Dcb (X, R), where b Kn ∈ ob Dctf (X, R/(λn+1 )) and n+1 ∼ Kn+1 ⊗L ) = Kn . R/(λn+2 ) R/(λ

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b We have RΓc (X, Kn ) ∈ ob Dperf (R/(λn+1 )) and

n+1 ∼ RΓc (X, Kn+1 ) ⊗L ) = RΓc (X, Kn ). R/(λn+2 ) R/(λ

Representing each RΓc (X, Kn ) by a bounded complex Ln of free R/(λn+1 )modules of finite rank. Then we have quasi-isomorphisms Ln+1 ⊗R/(λn+2 ) R/(λn+1 ) ∼ = Ln .

By 10.1.15, we can find a bounded complex L of free R-modules of finite rank and quasi-isomorphisms L/λn+1 L → Ln such that the diagrams L/λn+2 L → Ln+1 ↓ ↓ L/λn+1 L → Ln

commute up to homotopy. The morphism F ∗ on RΓc (X, Kn ) can be represented by a morphism of complexes Fn∗ on L/λn+1 L. By 10.1.11, we may assume So the family

∗ ⊗ idR/(λn+1 ) . Fn∗ = Fn+1

Tr F ∗ , RΓc (X, Kn ) = Tr(Fn∗ , L/λn+1 L)

defines an element in R = limn R/(λn+1 ), and this element is ←− Tr(F ∗ , RΓc (X, K)). Similarly, for any x ¯ ∈ X F , the family (Tr(Fx¯∗ , Kn¯x )) defines an element in R = limn R/(λn+1 ), and this element is Tr(Fx¯∗ , Kx¯ ). ←− By 10.4.4, we have X Tr(Fx¯∗ , Kn¯x ) = Tr(F ∗ , RΓc (X, Kn )) x ¯∈X F

for all n. So we have X

Tr(Fx¯∗ , Kx¯ ) = Tr(F ∗ , RΓc (X, Kn )).

x ¯∈X F

To prove 10.4.4, we first consider the special case where dim X ≤ 1, and then reduce the general case to the special case. Lemma 10.4.6. Let X be a scheme of finite type over an algebraically b closed field k, f : X → X a k-morphism, K ∈ ob Dctf (X, A), f ∗ : f ∗ K → f K a morphism, and X the set of fixed points of f in X(k). If dim X = 0, then we have X Tr(f ∗ , Kx¯ ) = Tr(f ∗ , RΓc (X, K)). x ¯∈X f

In particular, 10.4.4 holds if dim X0 = 0.

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Proof.

Use the fact that RΓc (X, K) =

M

Kx¯ .

x ¯∈X(k)

Lemma 10.4.7. Let X be an irreducible smooth projective curve over an algebraically closed field k, f : X → X a k-morphism with isolated fixed points, and U a dense open subset of X such that f −1 (U ) = U . For any fixed point x of f on X(k), the multiplicity of f at x is defined to be vx (f ∗ (πx ) − πx ), where vx is the valuation of the function field of X defined by x and πx is a uniformizer for this valuation. Suppose that fixed points of f in X − U have multiplicity 1. Let vU (f ) be the sum of multiplicities of fixed points of f in U . Then we have 2 X (−1)i Tr(f ∗ , Hci (U, Q` )) = vU (f ). i=0

Proof.

We have 2 X

(−1)i Tr(f ∗ , Hci (U, Q` ))

=

i=0 2 X i=0

(−1)i Tr(f ∗ , H i (X, Q` )) −

P2

i

∗

i

2 X i=0

(−1)i Tr(f ∗ , H i (X − U, Q` )).

By 8.6.8, i=0 (−1) Tr(f , H (X, Q` )) is the sum of multiplicities of fixed P2 points of f in X. By 10.4.6, i=0 (−1)i Tr(f ∗ , H i (X −U, Q`)) is the number of fixed points of f in X − U . Since fixed points of f in X − U have P2 multiplicity 1, i=0 (−1)i Tr(f ∗ , Hci (U, Q` )) is the sum of multiplicities of fixed points of f in U . Lemma 10.4.8. Let X0 be a smooth irreducible curve on Fq , and let F0 be a locally constant sheaf of A-modules on X0 such that there exists a galois etale covering space f : X00 → X0 with the property that f ∗ F0 is a constant sheaf associated to a finitely generated projective A-module. Then we have X Tr(Fx¯∗ , Fx¯ ) = Tr(F ∗ , RΓc (X, F )), x ¯∈X F

where F is the inverse image of F0 in X.

Proof. Let M = Γ(X00 , f ∗ F0 ) and let G be the galois group of the etale covering space f : X00 → X0 . Then M is a finitely generated projective A-module with a G-action. The sheaf f∗ Z/`n on X0 is a locally free sheaf

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of Z/`n [G]-modules. By 10.2.1, 7.4.7 and 7.8.1, we have RΓc (X, f∗ Z/`n ) ∈ b ob Dperf (Z/`n [G]). The canonical morphism adj : f∗ f ∗ F0 = f! f ∗ F0 → F0 is a morphism of G-sheaves, where F0 is provided with the trivial G-action. It induces an isomorphism ∼ =

(f∗ f ∗ F0 )G → F0 . Consider the morphism of sheaves M ⊗Z/`n f∗ Z/`n → f∗ f ∗ F0 induced by the morphism of presheaves M ⊗Z/`n (f∗ Z/`n )(U ) → (f∗ f ∗ F0 )(U ) = (f ∗ F )(U ×X0 X00 ), s ⊗ a 7→ a(s|U×X0 X00 )

for any etale X0 -scheme U , any a ∈ (f∗ Z/`n )(U ) = Z/`n (U ×X0 X00 ), and any s ∈ M = Γ(X00 , f ∗ F0 ). Since f ∗ F0 ∼ = M , this is an isomorphism of sheaves by 7.4.7. Note that this is an isomorphism of G-sheaves, where the action of G on M ⊗Z/`n (f∗ Z/`n ) is given by g(s ⊗ a) = gs ⊗ ga. We thus have F0 ∼ = (M ⊗Z/`n f∗ Z/`n )G ∼ = A ⊗A[G] (M ⊗Z/`n f∗ Z/`n ). So we have n L RΓc (X, F ) ∼ = A ⊗L A[G] (M ⊗Z/`n RΓc (X, f∗ Z/` )).

Representing RΓc (X, f∗ Z/`n ) by a perfect complex of Z/`n [G]-modules P . We then have RΓc (X, F ) ∼ = (M ⊗Z/`n P )G .

Through this isomorphism, the action of F ∗ on RΓc (X, F ) is induced by the action of F ∗ on RΓc (X, f∗ Z/`n ) ∼ = RΓc (X 0 , Z/`n ), where X 0 = X00 ⊗Fq F. Suppose that F has no fixed point in X(F) and let us prove Tr(F ∗ , RΓc (X, F )) = 0, that is, TrA (F ∗ , (M ⊗Z/`n P )G ) = 0. By 10.4.3, it suffices to prove TrA[G] (F ∗ , M ⊗Z/`n P ) = 0.

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By 10.4.2, it suffices to prove TrZ/`n [G] (F ∗ , P ) = 0. By 10.4.1, it suffices to prove θg (TrZ/`n [G] (F ∗ , P )) = 0 for any g ∈ G, that is,

θg (TrZ/`n [G] (F ∗ , RΓc (X 0 , Z/`n ))) = 0.

By 10.4.1, we have #C(g)θg (TrZ/`n [G] (F ∗ , RΓc (X 0 , Z/`n ))) = TrZ/`n ((F g −1 )∗ , RΓc (X 0 , Z/`n )). This formula is true for any n. It follows that θg (TrZ/`n [G] (F ∗ , RΓc (X 0 , Z/`n )) 2

≡

1 X (−1)i Tr((F g −1 )∗ , Hci (X 0 , Q` )) #C(g) i=0

mod `n .

To prove our assertion, it suffices to show 2 X

(−1)i Tr((F g −1 )∗ , Hci (X 0 , Q` )) = 0

i=0

0

if F has no fixed points in X(F). Let X 0 be the smooth compactification 0 0 0 ¯ be a fixed point of F g −1 in X (F), and let of X00 , let X = X 0 ⊗Fq F, let x 0 x0 ∈ X 0 be the image of x ¯ in X00 . Then we have g(x0 ) = x0 . Choose a uniformizer πx0 for OX 0 ,x0 . Then πx0 is also a uniformizer for OX 0 ,¯x . We 0 have vx¯ ((F g −1 )∗ (πx0 ) − πx0 ) = vx (g −1 (πx0 )q − πx0 ) = 1. So the multiplicity of F g −1 at x¯ is 1. Moreover, if x ¯ is a fixed point of F g −1 0 in X (F), then its image in X is a fixed point of F in X(F). But F has no fixed point in X(F). So F g −1 has no fixed point in X 0 (F). We thus have 2 X

(−1)i Tr((F g −1 )∗ , Hci (X 0 , Q` )) = 0

i=0

by 10.4.7. This proves Tr(F ∗ , RΓc (X, F )) = 0 under the assumption that F has no fixed point in X. In general, we have Tr(F ∗ , RΓc (X − X F , F |X−X F )) = 0

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by the above discussion. By 10.4.6, we have X Tr(F ∗ , RΓc (X F , F |X F )) = Tr(Fx¯∗ , Fx¯ ). x ¯∈X F

So we have Tr(F ∗ , RΓc (X, F )) = Tr(F ∗ , RΓc (X − X F , F |X−X F )) + Tr(F ∗ , RΓc (X F , F |X F )) X = Tr(Fx¯∗ , Fx¯ ). x ¯∈X F

Lemma 10.4.9. Let X0 be a smooth irreducible curve over Fq , and let b K ∈ ob Dctf (X0 , A). We have X Tr(Fx¯∗ , Kx¯ ) = Tr(F ∗ , RΓc (X, K)). x ¯∈X F

Proof. Represent K by a bounded complex of flat constructible sheaves of A-modules. By 5.8.1 (ii), we can find an open dense subset U0 of X0 , and an etale covering space U00 → U0 such that K|U00 is a complex of constant sheaves. By 10.4.8, we have X Tr(Fx¯∗ , Kx¯ ) = Tr(F ∗ , RΓc (U, K|U )), x ¯∈U F

where U = U0 ⊗Fq F. By 10.4.6, we have X Tr(Fx¯∗ , Kx¯ ) = Tr(F ∗ , RΓc (X − U, K|X−U )). x ¯∈(X−U)F

So we have Tr(F ∗ , RΓc (X, K)) = Tr(F ∗ , RΓc (U, K|U )) + Tr(F ∗ , RΓc (X − U, K|X−U )) X X = Tr(Fx¯∗ , Kx¯ ) + Tr(Fx¯∗ , Kx¯ ) x ¯∈U F

=

X

x ¯∈(X−U)F

Tr(Fx¯∗ , Kx¯ ).

x ¯∈X F

Proof of 10.4.4. For every locally closed subset Y of X, let X T1 (Y, K|Y ) = Tr(Fx¯∗ , Kx¯ ), x ¯∈Y F

T2 (Y, K|Y ) = Tr(F ∗ , RΓc (Y, K|Y )),

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and let S = {Y |Y is closed in X such that T1 (Y, K|Y ) 6= T2 (Y, K|Y )}. To prove 10.4.4, it suffices to show that S is empty. If S is not empty, then we can choose a minimal element Y in S . Let U be an affine open subset of Y . By the minimality of Y , we have T1 (Y − U, K|Y −U ) = T2 (Y − U, K|Y −U ). We have Tk (Y, K|Y ) = Tk (Y − U, K|Y −U ) + Tk (U, K|U ) (k = 1, 2). Since T1 (Y, K|Y ) 6= T2 (Y, K|Y ), we have T1 (U, K|U ) 6= T2 (U, K|U ).

Since U is affine, we can find a closed immersion i : U → Am Fq for some m. We have Tk (U, K|U ) = Tk (Am Fq , i∗ (K|U ))

(k = 1, 2).

b Denote i∗ (K|U ) also by K. We thus find a complex K ∈ ob Dctf (Am Fq , A) m m such that T1 (AFq , K) 6= T2 (AFq , K). m−1 be the projection. We have Let π : Am F q → AF q X Tr(Fx¯ , (Rπ! K)x¯ ) T1 (Am−1 Fq , Rπ! K) = (Fq ) x∈Am−1 Fq

=

X

(Fq ) x∈Am−1 Fq

Tr Fx¯ , RΓc π −1 (x) ⊗Fq F, K|π−1 (x)⊗Fq F .

By 10.4.9, we have Tr Fx¯ , RΓc π −1 (x) ⊗Fq F, K|π−1 (x)⊗Fq F =

So we have

T1 (Am−1 Fq , Rπ! K) =

X

(Fq ) x∈Am−1 Fq

X

Tr(Fy¯, Ky¯).

y∈π −1 (x)(Fq )

X

Tr(Fy¯ , Ky¯)

y∈π −1 (x)(Fq )

= T1 (Am Fq , K). On the other hand, we have ∗ m T2 (Am Fq , K) = Tr(F , RΓc (AF , K))

= Tr(F ∗ , RΓc (Am−1 , Rπ! K)) F = T2 (Am−1 Fq , Rπ! K).

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So we have m−1 Tk (Am Fq , K) = Tk (AFq , Rπ! K) (k = 1, 2). m−1 We thus have T1 (Am−1 Fq , Rπ! K) 6= T2 (AFq , Rπ! K). Denote Rπ! K also m−1 b by K. We thus find K ∈ ob Dctf (Am−1 Fq , A) such that T1 (AFq , K) 6= T2 (Am−1 Using this argument repeatedly, we can find K ∈ Fq , K). b ob Dctf (A1Fq , A) such that T1 (A1Fq , K) 6= T2 (A1Fq , K). This contradicts 10.4.9. So S is empty.

10.5

Grothendieck’s Formula of L-functions

([SGA 4 21 ] Rapport 3.) Let X0 be a compactifiable scheme over Fq , let X = X0 ⊗Fq F, let K0 ∈ ob Dcb (X0 , Q` ), and let K be the inverse image of K0 in X. Denote by |X0 | the set of closed point in X0 . For any x ∈ |X0 |, let N (x) be the number of elements of the residue field k(x), and let deg(x) = [k(x) : Fq ]. We have N (x) = q deg(x) . Let i : Spec k(x) → X0 be the closed immersion. The complex i∗ K0 is completely determined by the galois action of Gal(k(x)/k(x)) on Kx¯ . Let fx : Kx¯ → Kx¯ be the action of the Frobenius deg(x) substitution α 7→ αq in Gal(k(x)/k(x)).Define the L-function of K0 to be Y 1 , L(X0 , K0 , s) = −1 1 ¯ x∈|X0 | det 1 − N (x)s fx , Kx where

det 1 −

Y (−1)i 1 1 −1 −1 i f 1 − f , K = det , H (K ) . x ¯ x ¯ N (x)s x N (x)s x i

Making the change of variable t = q −s , we can also define the L-function as Y 1 . L(X0 , K0 , t) = deg(x) det(1 − t fx−1 , Kx¯ ) x∈|X | 0

Let F : X → X be the base change of frX0 : X0 → X0 . For any x ∈ |X0 | with deg(x) = n, any F-point of X with image x is a fixed point of F n . ∗ Iterating n times the Frobenius correspondence FK : F ∗ K → K induces a 0 n∗ n∗ morphism Fx¯ : Kx¯ → Kx¯ . By 10.3.5, we have Fx¯ = fx−1 . So we have Y 1 . L(X0 , K0 , t) = deg(x)∗ deg(x) Fx¯ , Kx¯ ) x∈|X0 | deg(1 − t

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Theorem 10.5.1 (Grothendieck). Notation as above, we have L(X0 , K0 , t) =

Y i

det(1 − F ∗ t, Hci (X, K))(−1)

i+1

as a formal power series in Q` [[t]]. Remark 10.5.2. Since Hci (X, K) is nonzero only for finitely many i, this formula shows that L(X0 , K0 , t) is a rational function. Proof of 10.5.1. Since both sides of the equation are formal power series with constant term 1, it suffices to show t

i+1 d d Y ln L(X0 , K0 , t) = t ln det(1 − F ∗ t, Hci (X, K))(−1) . dt dt i

Note that for any endomorphism φ : V → V on a finite dimensional Q` vector space V , we have t

∞ X d ln det(1 − φtk )−1 = kTr(φn )tkn . dt n=1

Indeed, let λ1 , . . . , λd be the eigenvalues of φ, where d = dim V . We have d d d Y k −1 t ln det(1 − φt ) = t ln (1 − λi tk )−1 dt dt i=1

=− =

=

d X d t ln(1 − λi tk ) dt i=1

d X kλi tk 1 − λi tk i=1 d X ∞ X

kλni tkn

i=1 n=1

= =

∞ X

n=1 ∞ X n=1

d X k( λni )tkn i=1

kTr(φn )tkn .

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So we have d ln L(X0 , K0 , t) dt Y d deg(x)∗ = t ln det(1 − tdeg(x) Fx¯ , Kx¯ )−1 dt t

x∈|X0 |

X

=

x∈|X0 |

t

d deg(x)∗ ln det(1 − tdeg(x) Fx¯ , Kx¯ )−1 dt

∞ X X

=

ndeg(x)∗

deg(x)Tr(Fx¯

, Kx¯ )tndeg(x)

x∈|X0 | n=1 ∞ X

=

X

deg(x)Tr(Fx¯m∗ , Kx¯ )tm

m=1 x ∈ |X0 | deg(x)|m

=

∞ X

X

m=1 x ¯∈X

Tr(Fx¯m∗ , Kx¯ )tm .

Fm

Here for the last equality, we use the fact that there are exactly deg(x) F-points fixed by F m with image x for any x ∈ |X0 | satisfying deg(x)|m. Applying 10.4.5 to the scheme X0 ⊗Fq Fqm , we get X X Tr(Fx¯m∗ , Kx¯ ) = (−1)i Tr(F ∗m , Hci (X, K)). x ¯∈X F m

i

So we have t

∞ X X d ln L(X0 , K0 , t) = (−1)i Tr(F ∗m , Hci (X, K))tm . dt m=1 i

On the other hand, we have i+1 d Y t ln det(1 − F ∗ t, Hci (X, K))(−1) dt i X d = (−1)i t ln det(1 − F ∗ t, Hci (X, K))−1 dt i =

∞ XX i

(−1)i Tr(F ∗m , Hci (X, K))tm .

m=1

So we have i+1 d d Y t ln L(X0 , K0 , t) = t ln det(1 − F ∗ t, Hci (X, K))(−1) . dt dt i

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Corollary 10.5.3. Let X0 and Y0 be Fq -schemes of finite type, and let f : X0 → Y0 be a compactifiable Fq -morphism. For any K0 ∈ ob Dcb (X0 , Q` ), we have L(X0 , K0 , t) = L(Y0 , Rf! K0 , t). Proof.

We have L(Y0 , Rf! K0 , t) Y = det(1 − tdeg(y) fy−1 , (Rf! K)y¯)−1 y∈|Y0 |

=

Y Y

y∈|Y0 | i

=

Y

det(1 − tdeg(y) fy−1 , Hci (f −1 (¯ y ), K|f −1 (¯y) ))(−1) Y

y∈|Y0 | x∈|f −1 (y)|

=

Y

x∈|X0 |

i+1

det(1 − tdeg(y)[k(x):k(y)] fx−1 , Kx¯ )

det(1 − tdeg(x) fx−1 , Kx¯ )

= L(X0 , K0 , t), where the third equality follows from 10.3.5 and 10.5.1 applied to the k(y)scheme f −1 (y). This proves our assertion.

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Bibliography

A. Grothendieck and J. Dieudonn´e, ´ ements de G´eom´etrie Alg´ebrique. El´ EGA I Le langage des sch´emas, Publ. Math. IHES 4 (1960). ´ EGA II Etude globale ´el´ementaire de quelques classes de morphismes, Ibid. 8 (1961). ´ EGA III Etude cohomologique des faisceaux coh´erents, Ibid. 11 (1961), 17 (1963). ´ EGA IV Etude locale des sch´emas et des morphismes de sch´emas, Ibid. 20 (1964), 24 (1965), 28 (1966), 32 (1967). A. Grothendieck et al. S´eminaire de G´eometrie Alg´ebrique. ´ SGA 1 Revˆetements Etales et Groupe Fondamental, Lecture Notes in Math. 224, Springer-Verlag (1971). SGA 4 (with M. Artin and J. L. Verdier) Th´eorie des Topos et Coho´ mologie Etale des Sch´emas, Lecture Notes in Math. 269, 270, 305, Springer-Verlag (1972–1973). ´ SGA 4 21 (by P. Deligne et al.) Cohomologie Etale, Lecture Notes in Math. 569, Springer-Verlag (1977). SGA 5 (with I. Bucur, C. Houzel, L. Illusie, J.-P. Jouanonou and J.-P. Serre) Cohomologie l-adique et Fonctions L, Lecture Notes in Math. 589, Springer-Verlag (1977). SGA 7 (with M. Raynaud, D. S. Rim, P. Deligne, and N. Katz) Groupes de Monodromie en G´eom´etrie Alg´ebrique, Lecture Notes in Math. 288, 340, Springer-Verlag (1972–1973). M. F. Artin, Algebraic approximation of structures over complete local rings, Publ. Math. IHES 36 (1969), 23–58. M. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra, AddisonWesley, Reading, Mass. (1969). A. Beilinson, J. Bernstein and P. Deligne. Faisceaux Pervers, in Analyse et Topologie sur les Espace Singuliers (I), Ast´erique 100 (1982). 607

etale

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S. Bosch, W. L¨ utkebohmert, and M. Raynaud. N´eron Models, Springer-Verlag (1990). P. Deligne. La conjecture de Weil I, Publ. Math. IHES 43 (1974), 273–307. P. Deligne. La conjecture de Weil II, Publ. Math. IHES 52 (1980), 137–252. L. Fu. Algebraic Geometry, Tsinghua University Press, 2006. E. Freitag and R. Kiehl. Etale Cohomology and the Weil Conjecture, SpringerVerlag (1988). A. Grothendieck. Sur quelques points d’alg`ebre homologique, Tˆ ohoku Math. J. 9 (1957), 119–221. R. Hartshorne.Algebraic Geometry, Springer-Verlag (1977). R. Hartshorne. Residue and Duality, Lecture Notes in Math. 20, Springer-Verlag (1966). H. Matsumura. Commutative algebra, W. A. Benjamin Co., New York (1970). J. Milne. Etale Cohomology, Princeton University Press 1980. G. Laumon. Transformation de Fourier, constantes d’´equations fontionnelles, et conjecture de Weil, Publ. Math. IHES 65 (1987), 131–210. J.-P. Serre. Local Fields, Sringer-Verlag (1979). (English translation of Corps Locaux, Hermann, Paris (1962)). J.-P. Serre. Cohomologie Galoisienne, Lecture Notes in Math. 5, Springer-Verlag (1964). J.-P. Serre. Linear Representations of Finite Groups, Springer-Verlag (1977). (English translation of Repr´esentations Lin´ aires des Groupes Finis, Hermann, Paris (1971)).

etale

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Index

E-sheaf, 558 G-torsor, 242 K-group, 571 L-function, 603 RF -acyclic, 284 λ-adic sheaf, 532 λ-torsion sheaf, 531 Q` -sheaves, 561 p-cohomological dimension, 161 ˇ Cech cohomology group, 172 ˇ Cech complex, 172

constant sheaf, 245 constructible set, 15 constructible sheaf, 247 corestriction homomorphism, 145, 155 cospecialization homomorphism, 390 cup product, 363 decomposition subgroup, 121 degree of a divisor, 317 degree of an invertible OX -module, 317 degree of inertia, 411 derived category, 275 descent datum for a quasi-coherent sheaf, 22 descent datum for a scheme, 38 dimension of a scheme, 311 direct image, 186 direct limit, 88 distinguished triangle, 267 division algebra, 232 divisor class group, 312

A-R λ-adic sheaf, 532 A-R category , 532 algebra of finite presentation, 48 algebraic geometric point, 378 Artin character, 574 Artin–Rees–Mittag–Leffler condition (ARML), 532 Artin–Schreier’s theory, 318 Cartier divisor, 314 Cartier divisor class group, 314 central K-algebra, 232 codimension, 311 cofinal, 90 cohomological dimension, 161 cohomology class associated to an algebraic cycle, 478, 482, 483 compactifiable morphism, 350 compactifiable scheme, 350 compactification, 351

effective Cartier divisor, 314 effective divisor, 312 equivalence of categories, 4 essentially etale homomorphism, 100 essentially surjective functor, 3 etale cohomology group, 210 etale covering, 178 etale covering space, 121 etale morphism, 66 609

etale

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Etale Cohomology Theory

etale neighborhood, 194 etale presheaf, 176 etale sheaf, 179 exact functor, 268 faithful functor, 3 faithfully flat module, 5 faithfully flat morphism, 20 final object, 90 finite cohomological dimension, 283 finite Tor-dimension, 295 flasque sheaf, 212 flat module, 1 flat morphism, 20 flat sheaf, 214 Frobenius correspondence, 586 Frobenius morphism, 585 fully faithful functor, 3 fundamental group, 128 galois etale covering space, 122 generalization of a geometric point, 196 generalization of a point, 17 geometric Frobenius correspondence, 589 geometric point, 121 germ, 194 Godement resolution, 219 Grothendieck topology, 192 group scheme, 240 henselian local ring, 95 henselization of a local ring, 102 higher direct image, 210 Hochschild–Serre spectral sequence, 144, 155, 501 induced G-module, 143, 155 inertia subgroup, 121, 414 inflation homomorphism, 142 inverse image, 186 inverse limit, 88 K¨ unneth formula, 363, 510 Kummer’s theory, 317

left derived functor, 281 lisse λ-adic sheaf, 532 lisse Q` -sheaf, 562 lisse E-sheaf, 559 localization of category, 273 locally acyclic morphism, 378 locally closed subset, 15 locally constant sheaf, 245 mapping cone, 271 Mittag–Leffler condition (ML), 532 module of relative differentials, 59 morphism locally of finite presentation, 49 morphism of finite presentation, 51 multiplicity of a fixed point, 494 nearby cycle functor, 505 noetherian sheaf, 249 normal basis, 165 null system, 532 perfect complex, 567 presheaf, 171 prime divisor, 311 principal Cartier divisor, 314 principal divisor, 312 pro-p-group, 151 profinite group, 133 projection formula, 361 proper base change theorem, 331 quasi-affine morphism, 40 quasi-affine scheme, 40 quasi-algebraically closed field, 168 quasi-compact morphism, 21 quasi-finite homomorphism, 41 quasi-finite morphism, 43 quasi-isomorphism, 274 quotient of a scheme, 117 radiciel morphism, 28 ramification index, 411 reduced norm, 238 refinement of an etale covering, 179 regular in codimension one, 311

etale

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611

Index

relative dimension, 73 relative Frobenius morphism, 586 representable functor, 59 restriction, 171 restriction homomorphism, 142 right derived functor, 280 sheaf associated to a presheaf, 179 sheaf of relative differentials, 63 sheaf represented by a scheme, 189 simple algebra, 232 simple module, 232 skyscraper sheaf, 319 smooth base change theorem, 391 smooth morphism, 73 specialization map, 198 specialization morphism, 196 specialization of a geometric point, 196 specialization of a point, 17 split complex, 219, 301 stalk, 193 strict p-cohomological dimension, 161 strict cohomological dimension, 161 strict henselization of a local ring, 111 strict localization of a local ring, 111 strict localization of a scheme, 196 strictly essentially etale homomorphism, 101 strictly henselian local ring, 95 strictly local ring, 95 strictly local trait, 434 strongly locally acyclic morphism, 378 support of a section, 201 support of a sheaf, 201 Swan character, 574 Sylow p-subgroup of a profinite group, 151

etale

tamely ramified extension, 412 thick subcategory of an abelian category, 271 totally ramified extension, 412 trace, 566 trait, 434 triangulated category, 267 uniformizer, 411 universally locally acyclic morphism, 378 universally strongly locally acyclic morphism, 380 unramified extension, 412 unramified morphism, 65 vanishing cycle functor, 506 variation, 506 way-out left functor, 303 way-out right functor, 303 weakly injective G-module, 143, 155 Weil divisor, 311 wild inertia subgroup, 416